
theorem Th14:
  for C being Category, I be Indexing of C for D being Categorial
  Category holds rng I is Subcategory of D iff D is TargetCat of I
proof
  let C be Category, I be Indexing of C;
  let D be Categorial Category;
  hereby
    assume
A1: rng I is Subcategory of D;
    thus D is TargetCat of I
    proof
      hereby
        let a be Object of C;
        I`1.a is Object of rng I by Def9;
        hence I`1.a is Object of D by A1,CAT_2:6;
      end;
      let b be Morphism of C;
      [[I`1.((the Source of C).b),I`1.((the Target of C).b)],I`2.b] is
      Morphism of rng I by Def9;
      hence thesis by A1,CAT_2:8;
    end;
  end;
  assume D is TargetCat of I;
  then reconsider T = D as TargetCat of I;
  rng I = Image (I-functor(C,T)) by Def12;
  hence thesis;
end;
