
theorem
  for X being set, C being Category holds Objs (X --> C) = X --> the
  carrier of C & Mphs (X --> C) = X --> the carrier' of C
proof
  let X be set, C be Category;
A2: dom Objs (X --> C) = dom (X --> C) by Def2;
  now
    let a be object;
    assume
A3: a in dom Objs (X --> C);
    then (X --> C).a = C by A2,FUNCOP_1:7;
    hence (Objs (X --> C)).a = the carrier of C by A2,A3,Def2;
  end;
  hence Objs (X --> C) = X --> the carrier of C by A2,FUNCOP_1:11;
A4: dom Mphs (X --> C) = dom (X --> C) by Def3;
  now
    let a be object;
    assume
A5: a in dom Mphs (X --> C);
    then (X --> C).a = C by A4,FUNCOP_1:7;
    hence (Mphs (X --> C)).a = the carrier' of C by A4,A5,Def3;
  end;
  hence thesis by A4,FUNCOP_1:11;
end;
