
theorem Th1:
  for f being one-to-one Function, y being object st rng f = {y}
  holds dom f = {f".y}
proof
  let f be one-to-one Function, y be object;
  assume A1: rng f = {y};
  then y in rng f by TARSKI:def 1;
  then consider x0 being object such that
    A2: x0 in dom f & f.x0 = y by FUNCT_1:def 3;
  for x being object holds x in dom f iff x = f".y
  proof
    let x be object;
    hereby
      assume A3: x in dom f;
      then f.x in rng f by FUNCT_1:3;
      then f.x = y by A1, TARSKI:def 1;
      hence x = f".y by A3, FUNCT_1:34;
    end;
    assume x = f".y;
    hence thesis by A2, FUNCT_1:34;
  end;
  hence thesis by TARSKI:def 1;
end;
