reserve G for _Graph;
reserve V for non empty set, E for Relation of V;

theorem
  rng E = rng the_Target_of createGraph(V,E)
proof
  now
    let w be object;
    hereby
      assume w in rng E;
      then consider v being object such that
        A1: [v,w] in E by XTUPLE_0:def 13;
      [v,w] DJoins v,w,createGraph(V,E) by A1, Th63;
      then A2: [v,w] in the_Edges_of createGraph(V,E) &
        (the_Target_of createGraph(V,E)).[v,w] = w by GLIB_000:def 14;
      then [v,w] in dom the_Target_of createGraph(V,E) by FUNCT_2:def 1;
      hence w in rng the_Target_of createGraph(V,E) by A2, FUNCT_1:3;
    end;
    assume w in rng the_Target_of createGraph(V,E);
    then consider e being object such that
      A3: e in dom the_Target_of createGraph(V,E) &
        (the_Target_of createGraph(V,E)).e = w by FUNCT_1:def 3;
    A4: e in the_Edges_of createGraph(V,E) by A3;
    set v = (the_Source_of createGraph(V,E)).e;
    e DJoins v,w,createGraph(V,E) by A3, GLIB_000:def 14;
    then A5: e = [v,w] by Th64;
    e in E by A4;
    hence w in rng E by A5, XTUPLE_0:def 13;
  end;
  hence thesis by TARSKI:2;
end;
