reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem
  G is connected iff G.allComponents() = { G | _GraphSelectors }
proof
  hereby
    assume A1: G is connected;
    then A2: { G | _GraphSelectors } c= G.allComponents()
      by Th192, ZFMISC_1:31;
    now
      let x be object;
      assume x in G.allComponents();
      then reconsider H = x as plain Component of G by Th189;
      A3: G == H by A1, GLIB_006:55;
      G == G | _GraphSelectors by GLIB_000:128;
      then H = G | _GraphSelectors by A3, GLIB_000:85, GLIB_009:44;
      hence x in { G | _GraphSelectors } by TARSKI:def 1;
    end;
    then G.allComponents() c= { G | _GraphSelectors } by TARSKI:def 3;
    hence G.allComponents() = { G | _GraphSelectors } by A2, XBOOLE_0:def 10;
  end;
  assume { G | _GraphSelectors } = G.allComponents();
  then G | _GraphSelectors in G.allComponents() by TARSKI:def 1;
  hence G is connected by Th192;
end;
