reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;

theorem
  for T,S being non empty set, f being Function of T,S, A,B being
  Subset-Family of S st A c= B holds f"A c= f"B
proof
  let T,S be non empty set;
  let f be Function of T,S;
  let A,B be Subset-Family of S;
  assume
A1: A c= B;
  let x be object;
  assume
A2: x in f"A;
  then reconsider x as Subset of T;
  ex C being Subset of S st C in B & x = f"C
  proof
    consider C being Subset of S such that
A3: C in A & x = f"C by A2,Def9;
    take C;
    thus thesis by A1,A3;
  end;
  hence thesis by Def9;
end;
