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 Y <> {} for y holds (ex x st x in X & x
  in P & y = f.x) implies y in f.:P
proof
  let f be Function of X,Y;
  assume Y <> {};
  then
A1: dom f = X by Def1;
  let y;
  given x such that
A2: x in X & x in P & y = f.x;
  thus thesis by A1,A2,FUNCT_1:def 6;
end;
