reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;

theorem Th5:
  for x being Element of X holds Im(f,x) = {f.x}
proof
  let x be Element of X;
  for y be object holds y in f.:{x} iff y = f.x
  proof
    let y be object;
    thus y in f.:{x} implies y = f.x
    proof
      assume y in f.:{x};
      then ex z being object st z in dom f & z in {x} & f.z = y
          by FUNCT_1:def 6;
      hence thesis by TARSKI:def 1;
    end;
    x in {x} by TARSKI:def 1;
    hence thesis by FUNCT_2:35;
  end;
  hence thesis by TARSKI:def 1;
end;
