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

theorem Th67:
  for F being PGraphMapping of G1, G2 st rng F_V = the_Vertices_of G2
  holds rng(SG2SGFunc(F) | G1.allSpanningSG()) c= G2.allSpanningSG()
proof
  let F be PGraphMapping of G1, G2;
  assume A1: rng F_V = the_Vertices_of G2;
  set f = SG2SGFunc(F) | G1.allSpanningSG();
  A3: dom f = G1.allSpanningSG() by FUNCT_2:def 1;
  now
    let y be object;
    assume A4: y in rng f;
    then consider x being object such that
      A5: x in dom f & f.x = y by FUNCT_1:def 3;
    consider H1 being Element of [#]G1.allSG() such that
      A6: x = H1 & H1 is spanning by A3, A5;
    reconsider H2 = y as Element of [#]G2.allSG() by A4;
    A7: (F|H1)_V = F_V | the_Vertices_of G1 by A6, GLIB_000:def 33
      .= F_V;
    H2 = SG2SGFunc(F).x by A5,FUNCT_1:47
      .= rng(F|H1) by A6, Def5;
    then H2 is spanning by A1, A7;
    hence y in G2.allSpanningSG();
  end;
  hence thesis by TARSKI:def 3;
end;
