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 Th22:
  R.:X = rng R & R"Y = dom R
proof
  now
    let y be object;
A1: now
      assume y in rng R;
      then consider x being object such that
A2:   [x,y] in R by XTUPLE_0:def 13;
      x in X by A2,ZFMISC_1:87;
      hence y in R.:X by A2,RELAT_1:def 13;
    end;
    now
      assume y in R.:X;
      then ex x being object st [x,y] in R & x in X by RELAT_1:def 13;
      hence y in rng R by XTUPLE_0:def 13;
    end;
    hence y in R.:X iff y in rng R by A1;
  end;
  hence R.:X = rng R by TARSKI:2;
  now
    let x be object;
A3: now
      assume x in dom R;
      then consider y being object such that
A4:   [x,y] in R by XTUPLE_0:def 12;
      y in Y by A4,ZFMISC_1:87;
      hence x in R"Y by A4,RELAT_1:def 14;
    end;
    now
      assume x in R"Y;
      then ex y being object st [x,y] in R & y in Y by RELAT_1:def 14;
      hence x in dom R by XTUPLE_0:def 12;
    end;
    hence x in R"Y iff x in dom R by A3;
  end;
  hence thesis by TARSKI:2;
end;
