let I be set ; :: thesis: for C being Category
for f being Morphism of C holds doms (I --> f) = I --> (dom f)
let C be Category; :: thesis: for f being Morphism of C holds doms (I --> f) = I --> (dom f)
let f be Morphism of C; :: thesis: doms (I --> f) = I --> (dom f)
set F = I --> f;
set F9 = I --> (dom f);

now :: thesis: for x being set st x in I holds
(doms (I --> f)) /. x = (I --> (dom f)) /. x

let x be set ; :: thesis: ( x in I implies (doms (I --> f)) /. x = (I --> (dom f)) /. x )
assume A1: x in I ; :: thesis: (doms (I --> f)) /. x = (I --> (dom f)) /. x
then ( (I --> f) /. x = f & (I --> (dom f)) /. x = dom f ) by Th2;
hence (doms (I --> f)) /. x = (I --> (dom f)) /. x by A1, Def1; :: thesis: verum

end;

hence doms (I --> f) = I --> (dom f) by Th1; :: thesis: verum