let C be Category; :: thesis: for O being non empty Subset of holds union { (Hom a,b) where a, b is Object of : ( a in O & b in O ) } is non empty Subset of
let O be non empty Subset of ; :: thesis: union { (Hom a,b) where a, b is Object of : ( a in O & b in O ) } is non empty Subset of
set H = { (Hom a,b) where a, b is Object of : ( a in O & b in O ) } ;
set M = union { (Hom a,b) where a, b is Object of : ( a in O & b in O ) } ;
A1: union { (Hom a,b) where a, b is Object of : ( a in O & b in O ) } c= the carrier' of C

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in union { (Hom a,b) where a, b is Object of : ( 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 : ( 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 : ( a in O & b in O ) } by TARSKI:def 4;
ex a, b being Object of 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

consider a being Element of O;
reconsider a = a as Object of ;
( id a in Hom a,a & Hom a,a in { (Hom a,b) where a, b is Object of : ( a in O & b in O ) } ) by CAT_1:55;
then id a in union { (Hom a,b) where a, b is Object of : ( 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 : ( a in O & b in O ) } ; :: thesis: verum

end;

hence union { (Hom a,b) where a, b is Object of : ( a in O & b in O ) } is non empty Subset of by A1; :: thesis: verum