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

theorem Th39:
  for f being Function of X,Y st Y = {} implies X = {} holds f"Y = X
proof
  let f be Function of X,Y;
  rng f /\ Y = rng f by XBOOLE_1:28;
  then
A1: f"Y = f"(rng f) by RELAT_1:133;
  assume Y <> {} or X = {};
  then dom f = X by Def1;
  hence thesis by A1,RELAT_1:134;
end;
