reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th21:
  for f being Function holds f is Cardinal-yielding iff
  for y being set st y in rng f holds y is Cardinal
proof
  let f be Function;
  hereby
    assume
A1: f is Cardinal-yielding;
    let y be set;
    assume y in rng f;
    then ex x being object st x in dom f & y = f.x by FUNCT_1:def 3;
    hence y is Cardinal by A1;
  end;
  assume
A2: for y being set st y in rng f holds y is Cardinal;
  let x be object;
  assume x in dom f;
  then f.x in rng f by FUNCT_1:def 3;
  hence thesis by A2;
end;
