reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem
  for f being Function of X,Y for g being Function of Y,X st X <> {} & Y
<> {} & rng f = Y & f is one-to-one &
  for y,x being object holds y in Y & g.y = x iff x in X
  & f.x = y holds g = f"
proof
  let f be Function of X,Y;
  let g be Function of Y,X;
  assume X <> {} & Y <> {};
  then dom f = X & dom g = Y by Def1;
  hence thesis by FUNCT_1:32;
end;
