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 = {} implies X = {} holds f is
  one-to-one iff
  for x1,x2 being object st x1 in X & x2 in X & f.x1 = f.x2 holds x1 = x2
proof
  let f be Function of X,Y;
  assume Y = {} implies X = {};
  then dom f = X by Def1;
  hence thesis;
end;
