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 Th3:
  dom {[x,y]} = {x} & rng {[x,y]} = {y}
proof
  set R = {[x,y]};
  for z being object holds z in dom R iff z in {x}
  proof let z be object;
    thus z in dom R implies z in {x}
    proof
      assume z in dom R;
      then consider b being object such that
A2:   [z,b] in R by XTUPLE_0:def 12;
      [z,b] = [x,y] by A2,TARSKI:def 1;
      then z=x by XTUPLE_0:1;
      hence thesis by TARSKI:def 1;
    end;
    assume z in {x};
    then z=x by TARSKI:def 1;
    then [z,y] in R by TARSKI:def 1;
    hence thesis by XTUPLE_0:def 12;
  end;
  hence dom R = {x} by TARSKI:2;
  for z being object holds z in rng R iff z in {y}
  proof let z be object;
    thus z in rng R implies z in {y}
    proof
      assume z in rng R;
      then consider a being object such that
A3:   [a,z] in R by XTUPLE_0:def 13;
      [a,z] = [x,y] by A3,TARSKI:def 1;
      then z = y by XTUPLE_0:1;
      hence thesis by TARSKI:def 1;
    end;
    assume z in {y};
    then z = y by TARSKI:def 1;
    then [x,z] in R by TARSKI:def 1;
    hence thesis by XTUPLE_0:def 13;
  end;
  hence thesis by TARSKI:2;
end;
