reserve X for set;

theorem Th5:
  for X,Y being non empty set
  for A being non empty Subset of X
  for x being Element of X
  st not x in A
  for f being Function of X,Y
  st f is one-to-one
  holds not f.x in (f .: A)
proof
  let X,Y be non empty set;
  let A be non empty Subset of X;
  let x be Element of X;
  assume A1: not x in A;
  let f be Function of X,Y;
  assume A2: f is one-to-one;
  A3: dom f = X by FUNCT_2:def 1;
  f.x in (f .: A) iff ex a being object st a in dom f & a in A & f.x = f.a
  by FUNCT_1:def 6;
  hence f.x in (f .: A) implies contradiction by A2,A3,A1,FUNCT_1:def 4;
end;
