
theorem
  for X, Y being set holds X <> {} & Y = {} iff product (X --> Y) = {}
proof
  let X, Y be set;
  hereby
    assume X <> {} & Y = {};
    then rng(X --> Y) = {{}} by FUNCOP_1:8;
    then {} in rng(X --> Y) by TARSKI:def 1;
    hence product(X --> Y) = {} by CARD_3:26;
  end;
  assume product (X --> Y) = {};
  hence thesis;
end;
