:: Functions and Their Basic Properties :: by Czes{\l}aw Byli\'nski :: :: Received March 3, 1989 :: Copyright (c) 1990-2021 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies RELAT_1, XBOOLE_0, ZFMISC_1, SUBSET_1, TARSKI, SETFAM_1, FUNCT_1; notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, SETFAM_1; constructors SETFAM_1, RELAT_1, XTUPLE_0; registrations XBOOLE_0, RELAT_1, ZFMISC_1; requirements SUBSET, BOOLE; begin reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object; definition let X be set; attr X is Function-like means :: FUNCT_1:def 1 for x,y1,y2 st [x,y1] in X & [x,y2] in X holds y1 = y2; end; registration cluster empty -> Function-like for set; end; registration cluster Function-like for Relation; end; definition mode Function is Function-like Relation; end; registration let a, b be object; cluster {[a,b]} -> Function-like; end; reserve f,g,g1,g2,h for Function, R,S for Relation; scheme :: FUNCT_1:sch 1 GraphFunc { A()->set,P[object,object] } : ex f st for x,y being object holds [x,y] in f iff x in A() & P[x,y] provided for x,y1,y2 being object st P[x,y1] & P[x,y2] holds y1 = y2; definition let f; let x be object; func f.x -> set means :: FUNCT_1:def 2 [x,it] in f if x in dom f otherwise it = {}; end; theorem :: FUNCT_1:1 [x,y] in f iff x in dom f & y = f.x; theorem :: FUNCT_1:2 dom f = dom g & (for x st x in dom f holds f.x = g.x) implies f = g; definition let f; redefine func rng f means :: FUNCT_1:def 3 for y being object holds y in it iff ex x being object st x in dom f & y = f.x; end; theorem :: FUNCT_1:3 x in dom f implies f.x in rng f; theorem :: FUNCT_1:4 dom f = {x} implies rng f = {f.x}; scheme :: FUNCT_1:sch 2 FuncEx { A()->set,P[object,object] } : ex f st dom f = A() & for x st x in A() holds P[x,f.x] provided for x,y1,y2 st x in A() & P[x,y1] & P[x,y2] holds y1 = y2 and for x st x in A() ex y st P[x,y]; scheme :: FUNCT_1:sch 3 Lambda { A() -> set,F(object) -> object } : ex f being Function st dom f = A() & for x st x in A() holds f.x = F(x); theorem :: FUNCT_1:5 X <> {} implies for y ex f st dom f = X & rng f = {y}; theorem :: FUNCT_1:6 (for f,g st dom f = X & dom g = X holds f = g) implies X = {}; theorem :: FUNCT_1:7 dom f = dom g & rng f = {y} & rng g = {y} implies f = g; theorem :: FUNCT_1:8 Y <> {} or X = {} implies ex f st X = dom f & rng f c= Y; theorem :: FUNCT_1:9 (for y st y in Y ex x st x in dom f & y = f.x) implies Y c= rng f; notation let f,g; synonym g*f for f*g; end; registration let f,g; cluster g*f -> Function-like; end; theorem :: FUNCT_1:10 for h st (for x holds x in dom h iff x in dom f & f.x in dom g) & (for x st x in dom h holds h.x = g.(f.x)) holds h = g*f; theorem :: FUNCT_1:11 x in dom(g*f) iff x in dom f & f.x in dom g; theorem :: FUNCT_1:12 x in dom(g*f) implies (g*f).x = g.(f.x); theorem :: FUNCT_1:13 x in dom f implies (g*f).x = g.(f.x); theorem :: FUNCT_1:14 z in rng(g*f) implies z in rng g; theorem :: FUNCT_1:15 dom(g*f) = dom f implies rng f c= dom g; theorem :: FUNCT_1:16 rng f c= Y & (for g,h st dom g = Y & dom h = Y & g*f = h*f holds g = h) implies Y = rng f; registration let X; cluster id X -> Function-like; end; theorem :: FUNCT_1:17 f = id X iff dom f = X & for x st x in X holds f.x = x; theorem :: FUNCT_1:18 x in X implies (id X).x = x; theorem :: FUNCT_1:19 dom(f*(id X)) = dom f /\ X; theorem :: FUNCT_1:20 x in dom f /\ X implies f.x = (f*(id X)).x; theorem :: FUNCT_1:21 x in dom((id Y)*f) iff x in dom f & f.x in Y; theorem :: FUNCT_1:22 (id X)*(id Y) = id(X /\ Y); theorem :: FUNCT_1:23 rng f = dom g & g*f = f implies g = id dom g; definition let f; attr f is one-to-one means :: FUNCT_1:def 4 for x1,x2 st x1 in dom f & x2 in dom f & f.x1 = f.x2 holds x1 = x2; end; theorem :: FUNCT_1:24 f is one-to-one & g is one-to-one implies g*f is one-to-one; theorem :: FUNCT_1:25 g*f is one-to-one & rng f c= dom g implies f is one-to-one; theorem :: FUNCT_1:26 g*f is one-to-one & rng f = dom g implies f is one-to-one & g is one-to-one; theorem :: FUNCT_1:27 f is one-to-one iff for g,h st rng g c= dom f & rng h c= dom f & dom g = dom h & f*g = f*h holds g = h; theorem :: FUNCT_1:28 dom f = X & dom g = X & rng g c= X & f is one-to-one & f*g = f implies g = id X; theorem :: FUNCT_1:29 rng(g*f) = rng g & g is one-to-one implies dom g c= rng f; registration let X be set; cluster id X -> one-to-one; end; ::$CT theorem :: FUNCT_1:31 (ex g st g*f = id dom f) implies f is one-to-one; registration cluster empty -> one-to-one for Function; end; registration cluster one-to-one for Function; end; registration let f be one-to-one Function; cluster f~ -> Function-like; end; definition let f; assume f is one-to-one; func f" -> Function equals :: FUNCT_1:def 5 f~; end; theorem :: FUNCT_1:32 f is one-to-one implies for g being Function holds g=f" iff dom g = rng f & for y,x holds y in rng f & x = g.y iff x in dom f & y = f.x; theorem :: FUNCT_1:33 f is one-to-one implies rng f = dom(f") & dom f = rng(f"); theorem :: FUNCT_1:34 f is one-to-one & x in dom f implies x = (f").(f.x) & x = (f"*f) .x; theorem :: FUNCT_1:35 f is one-to-one & y in rng f implies y = f.((f").y) & y = (f*f") .y; theorem :: FUNCT_1:36 f is one-to-one implies dom(f"*f) = dom f & rng(f"*f) = dom f; theorem :: FUNCT_1:37 f is one-to-one implies dom(f*f") = rng f & rng(f*f") = rng f; theorem :: FUNCT_1:38 f is one-to-one & dom f = rng g & rng f = dom g & (for x,y st x in dom f & y in dom g holds f.x = y iff g.y = x) implies g = f"; theorem :: FUNCT_1:39 f is one-to-one implies f"*f = id dom f & f*f" = id rng f; theorem :: FUNCT_1:40 f is one-to-one implies f" is one-to-one; registration let f be one-to-one Function; cluster f" -> one-to-one; let g be one-to-one Function; cluster g*f -> one-to-one; end; theorem :: FUNCT_1:41 f is one-to-one & rng f = dom g & g*f = id dom f implies g = f"; theorem :: FUNCT_1:42 f is one-to-one & rng g = dom f & f*g = id rng f implies g = f"; theorem :: FUNCT_1:43 f is one-to-one implies (f")" = f; theorem :: FUNCT_1:44 f is one-to-one & g is one-to-one implies (g*f)" = f"*g"; theorem :: FUNCT_1:45 (id X)" = id X; registration let f,X; cluster f|X -> Function-like; end; theorem :: FUNCT_1:46 dom g = dom f /\ X & (for x st x in dom g holds g.x = f.x) implies g = f|X; theorem :: FUNCT_1:47 x in dom(f|X) implies (f|X).x = f.x; theorem :: FUNCT_1:48 x in dom f /\ X implies (f|X).x = f.x; theorem :: FUNCT_1:49 x in X implies (f|X).x = f.x; theorem :: FUNCT_1:50 x in dom f & x in X implies f.x in rng(f|X); theorem :: FUNCT_1:51 X c= Y implies (f|X)|Y = f|X & (f|Y)|X = f|X; theorem :: FUNCT_1:52 f is one-to-one implies f|X is one-to-one; registration let Y,f; cluster Y|`f -> Function-like; end; theorem :: FUNCT_1:53 g = Y|`f iff (for x holds x in dom g iff x in dom f & f.x in Y) & for x st x in dom g holds g.x = f.x; theorem :: FUNCT_1:54 x in dom(Y|`f) iff x in dom f & f.x in Y; theorem :: FUNCT_1:55 x in dom(Y|`f) implies (Y|`f).x = f.x; theorem :: FUNCT_1:56 dom(Y|`f) c= dom f; theorem :: FUNCT_1:57 X c= Y implies Y|`(X|`f) = X|`f & X|`(Y|`f) = X|`f; theorem :: FUNCT_1:58 f is one-to-one implies Y|`f is one-to-one; definition let f,X; redefine func f.:X means :: FUNCT_1:def 6 for y being object holds y in it iff ex x being object st x in dom f & x in X & y = f.x; end; theorem :: FUNCT_1:59 x in dom f implies Im(f,x) = {f.x}; theorem :: FUNCT_1:60 x1 in dom f & x2 in dom f implies f.:{x1,x2} = {f.x1,f.x2}; theorem :: FUNCT_1:61 (Y|`f).:X c= f.:X; theorem :: FUNCT_1:62 f is one-to-one implies f.:(X1 /\ X2) = f.:X1 /\ f.:X2; theorem :: FUNCT_1:63 (for X1,X2 holds f.:(X1 /\ X2) = f.:X1 /\ f.:X2) implies f is one-to-one; theorem :: FUNCT_1:64 f is one-to-one implies f.:(X1 \ X2) = f.:X1 \ f.:X2; theorem :: FUNCT_1:65 (for X1,X2 holds f.:(X1 \ X2) = f.:X1 \ f.:X2) implies f is one-to-one; theorem :: FUNCT_1:66 X misses Y & f is one-to-one implies f.:X misses f.:Y; theorem :: FUNCT_1:67 (Y|`f).:X = Y /\ f.:X; definition let f,Y; redefine func f"Y means :: FUNCT_1:def 7 for x holds x in it iff x in dom f & f.x in Y; end; theorem :: FUNCT_1:68 f"(Y1 /\ Y2) = f"Y1 /\ f"Y2; theorem :: FUNCT_1:69 f"(Y1 \ Y2) = f"Y1 \ f"Y2; theorem :: FUNCT_1:70 (R|X)"Y = X /\ (R"Y); theorem :: FUNCT_1:71 for f being Function, A,B being set st A misses B holds f"A misses f"B; theorem :: FUNCT_1:72 y in rng R iff R"{y} <> {}; theorem :: FUNCT_1:73 (for y st y in Y holds R"{y} <> {}) implies Y c= rng R; theorem :: FUNCT_1:74 (for y st y in rng f ex x st f"{y} = {x}) iff f is one-to-one; theorem :: FUNCT_1:75 f.:(f"Y) c= Y; theorem :: FUNCT_1:76 X c= dom R implies X c= R"(R.:X); theorem :: FUNCT_1:77 Y c= rng f implies f.:(f"Y) = Y; theorem :: FUNCT_1:78 f.:(f"Y) = Y /\ f.:(dom f); theorem :: FUNCT_1:79 f.:(X /\ f"Y) c= (f.:X) /\ Y; theorem :: FUNCT_1:80 f.:(X /\ f"Y) = (f.:X) /\ Y; theorem :: FUNCT_1:81 X /\ R"Y c= R"(R.:X /\ Y); theorem :: FUNCT_1:82 f is one-to-one implies f"(f.:X) c= X; theorem :: FUNCT_1:83 (for X holds f"(f.:X) c= X) implies f is one-to-one; theorem :: FUNCT_1:84 f is one-to-one implies f.:X = (f")"X; theorem :: FUNCT_1:85 f is one-to-one implies f"Y = (f").:Y; :: SUPLEMENT theorem :: FUNCT_1:86 Y = rng f & dom g = Y & dom h = Y & g*f = h*f implies g = h; theorem :: FUNCT_1:87 f.:X1 c= f.:X2 & X1 c= dom f & f is one-to-one implies X1 c= X2; theorem :: FUNCT_1:88 f"Y1 c= f"Y2 & Y1 c= rng f implies Y1 c= Y2; theorem :: FUNCT_1:89 f is one-to-one iff for y ex x st f"{y} c= {x}; theorem :: FUNCT_1:90 rng R c= dom S implies R"X c= (R*S)"(S.:X); theorem :: FUNCT_1:91 for f being Function st f " X = f " Y & X c= rng f & Y c= rng f holds X = Y; begin :: Addenda :: from BORSUK_1 reserve e,u for object, A for Subset of X; theorem :: FUNCT_1:92 (id X).:A = A; :: from PBOOLE definition let f be Function; redefine attr f is empty-yielding means :: FUNCT_1:def 8 for x st x in dom f holds f.x is empty; end; :: from UNIALG_1 definition let F be Function; redefine attr F is non-empty means :: FUNCT_1:def 9 for n being object st n in dom F holds F.n is non empty; end; :: new, 2004.08.04 registration cluster non-empty for Function; end; :: from MSUALG_2 scheme :: FUNCT_1:sch 4 LambdaB { D()->non empty set, F(object)->object } : ex f be Function st dom f = D() & for d be Element of D() holds f.d = F(d); :: from PUA2MSS1, 2005.08.22, A.T. registration let f be non-empty Function; cluster rng f -> with_non-empty_elements; end; :: from SEQM_3, 2005.12.17, A.T. definition let f be Function; attr f is constant means :: FUNCT_1:def 10 x in dom f & y in dom f implies f.x = f.y; end; theorem :: FUNCT_1:93 for A,B being set, f being Function st A c= dom f & f.:A c= B holds A c= f"B; :: moved from MSAFREE3:1, AG 1.04.2006 theorem :: FUNCT_1:94 for f being Function st X c= dom f & f is one-to-one holds f"(f.:X) = X; :: added, AK 5.02.2007 definition let f,g; redefine pred f = g means :: FUNCT_1:def 11 dom f = dom g & for x st x in dom f holds f.x = g.x; end; :: missing, 2007.03.09, A.T. registration cluster non-empty non empty for Function; end; :: from PRVECT_1, 2007.03.09, A.T. registration let a be non-empty non empty Function; let i be Element of dom a; cluster a.i -> non empty; end; :: missing, 2007.04.13, A.T. registration let f be Function; cluster -> Function-like for Subset of f; end; :: from SCMFSA6A, 2007.07.23, A.T. theorem :: FUNCT_1:95 for f,g being Function, D being set st D c= dom f & D c= dom g holds f | D = g | D iff for x being set st x in D holds f.x = g.x; :: from SCMBSORT, 2007.07.26, A.T. theorem :: FUNCT_1:96 for f,g being Function, X being set st dom f = dom g & (for x being set st x in X holds f.x = g.x) holds f|X = g|X; :: missing, 2007.10.28, A.T. theorem :: FUNCT_1:97 rng(f|{X}) c= {f.X}; theorem :: FUNCT_1:98 X in dom f implies rng(f|{X}) ={f.X}; :: from RFUNCT_1, 2008.09.04, A.T. registration cluster empty -> constant for Function; end; :: from WAYBEL35, 2008.08.04, A.T. registration let f be constant Function; cluster rng f -> trivial; end; registration cluster non constant for Function; end; registration let f be non constant Function; cluster rng f -> non trivial; end; registration cluster non constant -> non trivial for Function; end; registration cluster trivial -> constant for Function; end; :: from RFUNCT_2, 2008.09.14, A.T. theorem :: FUNCT_1:99 for F,G be Function, X holds (G|(F.:X))*(F|X) = (G*F)|X; theorem :: FUNCT_1:100 for F,G be Function, X,X1 holds (G|X1)*(F|X) = (G*F)|(X /\ (F"X1)); theorem :: FUNCT_1:101 for F,G be Function,X holds X c= dom (G*F) iff X c= dom F & F.:X c= dom G; :: from YELLOW_6, 2008.12.26, A.T. definition let f be Function; assume f is non empty constant; func the_value_of f -> object means :: FUNCT_1:def 12 ex x being set st x in dom f & it = f.x; end; :: from QC_LANG4, 2009.01.23, A.T registration let X,Y; cluster X-defined Y-valued for Function; end; theorem :: FUNCT_1:102 for X being set, f being X-valued Function for x being set st x in dom f holds f.x in X; :: from FRAENKEL, 2009.05.06, A.K. definition let IT be set; attr IT is functional means :: FUNCT_1:def 13 for x being object st x in IT holds x is Function; end; registration cluster empty -> functional for set; let f be Function; cluster { f } -> functional; let g be Function; cluster { f,g } -> functional; end; registration cluster non empty functional for set; end; registration let P be functional set; cluster -> Function-like Relation-like for Element of P; end; registration let A be functional set; cluster -> functional for Subset of A; end; :: new, 2009.09.30, A.T. definition let g,f be Function; attr f is g-compatible means :: FUNCT_1:def 14 x in dom f implies f.x in g.x; end; theorem :: FUNCT_1:103 f is g-compatible & dom f = dom g implies g is non-empty; theorem :: FUNCT_1:104 {} is f-compatible; registration let I be set, f be Function; cluster empty I-defined f-compatible for Function; end; registration let X be set; let f be Function, g be f-compatible Function; cluster g|X -> f-compatible; end; registration let I be set; cluster non-empty I-defined for Function; end; theorem :: FUNCT_1:105 for g being f-compatible Function holds dom g c= dom f; registration let X; let f be X-defined Function; cluster f-compatible -> X-defined for Function; end; theorem :: FUNCT_1:106 for f being X-valued Function st x in dom f holds f.x is Element of X; :: from JGRAPH_6, 2010.03.15, A.T. theorem :: FUNCT_1:107 for f being Function,A being set st f is one-to-one & A c= dom f holds f".:(f.:A)=A; registration let A be functional set, x be object; let F be A-valued Function; cluster F.x -> Function-like Relation-like; end; :: missing, 2011.03.06, A.T. theorem :: FUNCT_1:108 x in X & x in dom f implies f.x in f.:X; theorem :: FUNCT_1:109 X <> {} & X c= dom f implies f.:X <> {}; registration let f be non trivial Function; cluster dom f -> non trivial; end; :: from HAHNBAN, 2011.04.26, A.T. theorem :: FUNCT_1:110 for B being non empty functional set, f being Function st f = union B holds dom f = union the set of all dom g where g is Element of B & rng f = union the set of all rng g where g is Element of B; scheme :: FUNCT_1:sch 5 LambdaS { A() -> set,F(object) -> object } : ex f being Function st dom f = A() & for X st X in A() holds f.X = F(X); theorem :: FUNCT_1:111 :: WELLORD2:28 for M being set st for X st X in M holds X <> {} ex f being Function st dom f = M & for X st X in M holds f.X in X; scheme :: FUNCT_1:sch 6 NonUniqBoundFuncEx { X() -> set, Y() -> set, P[object,object] }: ex f being Function st dom f = X() & rng f c= Y() & for x being object st x in X() holds P[x,f.x] provided for x being object st x in X() ex y being object st y in Y() & P[x,y]; registration let f be empty-yielding Function; let x; cluster f.x -> empty; end; :: from PNPROC_1, 2012.02.20, A.T. theorem :: FUNCT_1:112 for f,g,h being Function st f c= h & g c= h & f misses g holds dom f misses dom g; theorem :: FUNCT_1:113 for Y being set, f being Function holds Y|`f = f|(f"Y); registration let X be set; let x be Element of X; reduce (id X).x to x; end; theorem :: FUNCT_1:114 rng f c= rng g implies for x being object st x in dom f ex y being object st y in dom g & f.x = g. y;