theorem
  for y being object holds y in rng R iff
   ex x being Element of D st [x,y] in R
proof
  let y be object;
  thus y in rng R implies ex x being Element of D st [x,y] in R
  proof
    assume y in rng R;
    then consider x being object such that
A1: [x,y] in R by XTUPLE_0:def 13;
    reconsider a = x as Element of D by A1,ZFMISC_1:87;
    take a;
    thus thesis by A1;
  end;
  thus thesis by XTUPLE_0:def 13;
end;
