theorem Th55:
  for T, S being 1-sorted, f being Function of T,S, P1 being
  Subset of S st rng f = [#]S & f is one-to-one holds f"P1 = (f").:P1
proof
  let T, S be 1-sorted, f be Function of T, S, P1 be Subset of S;
  assume rng f = [#]S;
  then
A1: f is onto by FUNCT_2:def 3;
  assume
A2: f is one-to-one;
  f"P1 = ((f qua Function)").:P1 by A2,FUNCT_1:85;
  hence thesis by A1,A2,Def4;
end;
