let C be category; :: thesis: AllMono (AllMono C) = AllMono C
A1: ( the carrier of (AllMono (AllMono C)) = the carrier of (AllMono C) & the carrier of (AllMono C) = the carrier of C ) by Def1;
A2: the Arrows of (AllMono (AllMono C)) cc= the Arrows of (AllMono C) by Def1;
now :: thesis: for i being object st i in [: the carrier of C, the carrier of C:] holds
the Arrows of (AllMono (AllMono C)) . i = the Arrows of (AllMono C) . i
let i be object ; :: thesis: ( i in [: the carrier of C, the carrier of C:] implies the Arrows of (AllMono (AllMono C)) . i = the Arrows of (AllMono C) . i )
assume A3: i in [: the carrier of C, the carrier of C:] ; :: thesis: the Arrows of (AllMono (AllMono C)) . i = the Arrows of (AllMono C) . i
then consider o1, o2 being object such that
A4: ( o1 in the carrier of C & o2 in the carrier of C ) and
A5: i = [o1,o2] by ZFMISC_1:84;
reconsider o1 = o1, o2 = o2 as Object of (AllMono C) by A4, Def1;
thus the Arrows of (AllMono (AllMono C)) . i = the Arrows of (AllMono C) . i :: thesis: verum
proof
thus the Arrows of (AllMono (AllMono C)) . i c= the Arrows of (AllMono C) . i by A1, A2, A3; :: according to XBOOLE_0:def 10 :: thesis: the Arrows of (AllMono C) . i c= the Arrows of (AllMono (AllMono C)) . i
let n be object ; :: according to TARSKI:def 3 :: thesis: ( not n in the Arrows of (AllMono C) . i or n in the Arrows of (AllMono (AllMono C)) . i )
assume A6: n in the Arrows of (AllMono C) . i ; :: thesis: n in the Arrows of (AllMono (AllMono C)) . i
then reconsider n1 = n as Morphism of o1,o2 by A5;
n1 is mono by A5, A6, Th50;
then n in the Arrows of (AllMono (AllMono C)) . (o1,o2) by A5, A6, Def1;
hence n in the Arrows of (AllMono (AllMono C)) . i by A5; :: thesis: verum
end;
end;
hence AllMono (AllMono C) = AllMono C by A1, ALTCAT_2:25, PBOOLE:3; :: thesis: verum