reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object;
reserve P,R for Relation of X,Y;

theorem
  (for y being object st y in Y
    ex x being object st [x,y] in R) iff rng R = Y
proof
  thus (for y being object st y in Y
        ex x being object st [x,y] in R) implies rng R = Y
  proof
    assume
A1: for y being object st y in Y ex x being object st [x,y] in R;
    now
      let y be object;
      now
        assume y in Y;
        then ex x being object st [x,y] in R by A1;
        hence y in rng R by XTUPLE_0:def 13;
      end;
      hence y in rng R iff y in Y;
    end;
    hence rng R = Y by TARSKI:2;
  end;
  thus thesis by XTUPLE_0:def 13;
end;
