reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem
  for f being Function of X,Y st for y being object st y in Y
   ex x being object st x in X & y = f.x
  holds rng f = Y
proof
  let f be Function of X,Y such that
A1: for y being object st y in Y ex x being object st x in X & y = f.x;
  per cases;
  suppose Y = {};
   hence thesis;
  end;
  suppose A2: Y <> {};
  for y being object holds y in rng f iff y in Y
  proof let y be object;
    dom f = X by A2,Def1;
    then y in rng f iff ex x being object st x in X & y = f.x
   by FUNCT_1:def 3;
    hence thesis by A1;
  end;
  hence thesis by TARSKI:2;
  end;
end;
