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

theorem Th32:
  for F being PGraphMapping of G1, G2 st F is onto
  holds rng SG2SGFunc(F) = G2.allSG()
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is onto;
  now
    let x be object;
    assume x in G2.allSG();
    then reconsider H2 = x as plain Subgraph of G2 by Th1;
    set H1 = the plain inducedSubgraph of
      G1, F_V"the_Vertices_of H2, F_E"the_Edges_of H2;
    H1 in G1.allSG() by Th1;
    then A2: H1 in dom SG2SGFunc(F) by FUNCT_2:def 1;
    rng F == G2 by A1, GLIB_010:56;
    then H2 is Subgraph of rng F by GLIB_000:91;
    then H2 = rng(F | H1) by A1, GLIB_009:44, GLIBPRE1:99
      .= SG2SGFunc(F).H1 by Def5;
    hence x in rng SG2SGFunc(F) by A2, FUNCT_1:def 3;
  end;
  then G2.allSG() c= rng SG2SGFunc(F) by TARSKI:def 3;
  hence thesis by XBOOLE_0:def 10;
end;
