
theorem
  for G1, G2 being _Graph
  holds {G1}, {G2} are_isomorphic iff G2 is G1-isomorphic
proof
  let G1, G2 be _Graph;
  hereby
    assume {G1}, {G2} are_isomorphic;
    then consider f being one-to-one Function such that
      A1: dom f = {G1} & rng f = {G2} and
      A2: for G being _Graph st G in {G1} holds f.G is G-isomorphic _Graph;
    A3: f = {G1} --> G2 by A1, FUNCOP_1:9;
    G1 in {G1} by TARSKI:def 1;
    then f.G1 is G1-isomorphic _Graph & f.G1 = G2 by A2, A3, FUNCOP_1:7;
    hence G2 is G1-isomorphic;
  end;
  assume A4: G2 is G1-isomorphic;
  reconsider f = G1 .--> G2 as one-to-one Function;
  take f;
  f = {G1} --> G2 by FUNCOP_1:def 9;
  hence dom f = {G1} & rng f = {G2} by FUNCOP_1:8;
  let G be _Graph;
  assume G in {G1};
  then G = G1 by TARSKI:def 1;
  hence f.G is G-isomorphic _Graph by A4, FUNCOP_1:72;
end;
