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 x being object st x in X
    ex y being object st [x,y] in R) iff dom R = X
proof
  thus (for x being object st x in X ex y being object st [x,y] in R)
     implies dom R = X
  proof
    assume
A1: for x being object st x in X ex y being object st [x,y] in R;
    now
      let x be object;
      now
        assume x in X;
        then ex y being object st [x,y] in R by A1;
        hence x in dom R by XTUPLE_0:def 12;
      end;
      hence x in dom R iff x in X;
    end;
    hence dom R = X by TARSKI:2;
  end;
  thus thesis by XTUPLE_0:def 12;
end;
