reserve a,x,y for object, A,B for set,
  l,m,n for Nat;

theorem
  for X being set, Y being non empty set, f being Function of X,Y st f
is one-to-one for B being Subset of X, C being Subset of Y st C c= f.:B holds f
  "C c= B
proof
  let X be set, Y be non empty set, f be Function of X,Y such that
A1: f is one-to-one;
  let B be Subset of X, C be Subset of Y;
  assume C c= f.:B;
  then
A2: f"C c= f"(f.:B) by RELAT_1:143;
  f"(f.:B) c= B by A1,FUNCT_1:82;
  hence thesis by A2;
end;
