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

theorem Th25:
  for f being Function of X,Y st f is one-to-one & rng f = Y holds
  f" is Function of Y,X
proof
  let f be Function of X,Y;
  assume that
A1: f is one-to-one and
A2: rng f = Y;
A3: rng(f") c= X
  proof
    let x be object;
    assume x in rng(f");
    then x in dom f by A1,FUNCT_1:33;
    hence thesis;
  end;
  dom(f") = Y by A1,A2,FUNCT_1:33;
  then reconsider R = f" as Relation of Y,X by A3,RELSET_1:4;
  R is quasi_total
  proof
    per cases;
    case X <> {};
      thus thesis by A1,A2,FUNCT_1:33;
    end;
    case X = {};
      then rng f = {};
      then dom(f") = {} by A1,FUNCT_1:32;
      hence thesis;
    end;
  end;
  hence thesis;
end;
