
theorem Th1:
  for X,Y being non empty set, f being Function of X,Y holds for A
  being Subset of X st for x1,x2 being Element of X holds x1 in A & f.x1=f.x2
  implies x2 in A holds f"(f.:A) = A
proof
  let X,Y be non empty set;
  let f be Function of X,Y;
  let A be Subset of X;
  assume
A1: for x1,x2 being Element of X holds x1 in A & f.x1=f.x2 implies x2 in A;
  for x being object st x in f"(f.:A) holds x in A
  proof
    let x be object;
    assume
A2: x in f"(f.:A);
    then f.x in f.:A by FUNCT_1:def 7;
    then ex x0 being object st x0 in X & x0 in A & f.x = f.x0 by FUNCT_2:64;
    hence thesis by A1,A2;
  end;
  then A c= f"(f.:A) & f"(f.:A) c= A by FUNCT_2:42,TARSKI:def 3;
  hence thesis by XBOOLE_0:def 10;
end;
