let V, C be set ; :: thesis: for a being finite Element of PFuncs (V,C) holds {a} in SubstitutionSet (V,C)
let a be finite Element of PFuncs (V,C); :: thesis: {a} in SubstitutionSet (V,C)
A1: for s, t being Element of PFuncs (V,C) st s in {a} & t in {a} & s c= t holds
s = t
proof
let s, t be Element of PFuncs (V,C); :: thesis: ( s in {a} & t in {a} & s c= t implies s = t )
assume that
A2: s in {a} and
A3: t in {a} and
s c= t ; :: thesis: s = t
s = a by A2, TARSKI:def 1;
hence s = t by A3, TARSKI:def 1; :: thesis: verum
end;
for u being set st u in {a} holds
u is finite ;
then {.a.} in { A where A is Element of Fin (PFuncs (V,C)) : ( ( for u being set st u in A holds
u is finite ) & ( for s, t being Element of PFuncs (V,C) st s in A & t in A & s c= t holds
s = t ) ) } by A1;
hence {a} in SubstitutionSet (V,C) by SUBSTLAT:def 1; :: thesis: verum