let C be Category; :: thesis: for O being non empty Subset of the carrier of C holds union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } is non empty Subset of the carrier' of C
let O be non empty Subset of the carrier of C; :: thesis: union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } is non empty Subset of the carrier' of C
set H = { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } ;
set M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } ;
A1: union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } c= the carrier' of C
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } or x in the carrier' of C )
assume x in union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } ; :: thesis: x in the carrier' of C
then consider X being set such that
A2: x in X and
A3: X in { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } by TARSKI:def 4;
ex a, b being Object of C st
( X = Hom (a,b) & a in O & b in O ) by A3;
hence x in the carrier' of C by A2; :: thesis: verum
end;
now :: thesis: ex f being set st f in union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) }
set a = the Element of O;
reconsider a = the Element of O as Object of C ;
( id a in Hom (a,a) & Hom (a,a) in { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } ) by CAT_1:27;
then id a in union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } by TARSKI:def 4;
hence ex f being set st f in union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } ; :: thesis: verum
end;
hence union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } is non empty Subset of the carrier' of C by A1; :: thesis: verum