theorem
  Y c= rng f implies f.:(f"Y) = Y
proof
  assume
A1: Y c= rng f;
  thus f.:(f"Y) c= Y by Th74;
  let y be object;
  assume
A2: y in Y;
  then consider x being object such that
A3: x in dom f & y = f.x by A1,Def3;
  x in f"Y by A2,A3,Def7;
  hence thesis by A3,Def6;
end;
