reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  dom {[a,b],[x,y]} = {a,x} & rng {[a,b],[x,y]} = {b,y}
proof
  set R = {[a,b],[x,y]};
  thus dom R = {a,x}
  proof
    thus dom R c= {a,x}
    proof
      let z be object;
      assume z in dom R;
      then consider c being object such that
A2:   [z,c] in R by XTUPLE_0:def 12;
      [z,c] = [a,b] or [z,c] = [x,y] by A2,TARSKI:def 2;
      then z = a or z = x by XTUPLE_0:1;
      hence thesis by TARSKI:def 2;
    end;
    let z be object;
    assume z in {a,x};
    then z = a or z = x by TARSKI:def 2;
    then [z,b] in R or [z,y] in R by TARSKI:def 2;
    hence thesis by XTUPLE_0:def 12;
  end;
  thus rng R c= {b,y}
  proof
    let z be object;
    assume z in rng R;
    then consider d being object such that
A3: [d,z] in R by XTUPLE_0:def 13;
    [d,z] = [a,b] or [d,z] = [x,y] by A3,TARSKI:def 2;
    then z = b or z = y by XTUPLE_0:1;
    hence thesis by TARSKI:def 2;
  end;
  let z be object;
  assume z in {b,y};
  then z = b or z = y by TARSKI:def 2;
  then [a,z] in R or [x,z] in R by TARSKI:def 2;
  hence thesis by XTUPLE_0:def 13;
end;
