
theorem Th13:
  for X, Y being non empty set, y being Element of Y
  holds X --> y in product (X --> Y)
proof
  let X, Y be non empty set, y be Element of Y;
  set f = X --> y;
  A1: dom f = X
    .= dom (X --> Y);
  for x being object st x in dom (X --> Y) holds f.x in (X --> Y).x
  proof
    let x be object;
    assume A2: x in dom (X --> Y);
    then A3: (X --> Y).x = Y by FUNCOP_1:7;
    f.x = y by A2, FUNCOP_1:7;
    hence thesis by A3;
  end;
  hence thesis by A1, CARD_3:def 5;
end;
