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

theorem Th68:
  for F being PGraphMapping of G1, G2 st F is onto & F_V is one-to-one total
  holds rng(SG2SGFunc(F) | G1.allSpanningSG()) = G2.allSpanningSG()
proof
  let F be PGraphMapping of G1, G2;
  set f = SG2SGFunc(F) | G1.allSpanningSG();
  assume A1: F is onto & F_V is one-to-one total;
  then rng F_V = the_Vertices_of G2 by GLIB_010:def 12;
  then A2: rng f c= G2.allSpanningSG() by Th67;
  now
    let y be object;
    assume y in G2.allSpanningSG();
    then consider H2 being Element of [#]G2.allSG() such that
      A3: y = H2 & H2 is spanning;
    H2 in G2.allSG();
    then H2 in rng SG2SGFunc(F) by A1, Th32;
    then consider x being object such that
      A4: x in dom SG2SGFunc(F) & (SG2SGFunc F).x = H2
      by FUNCT_1:def 3;
    reconsider H1 = x as plain Subgraph of G1 by A4, Th1;
    dom F_V = the_Vertices_of G1 by A1, PARTFUN1:def 2;
    then A5: the_Vertices_of H1 c= dom F_V;
    then dom((F|H1)_V) <> {} by RELAT_1:62;
    then A6: F|H1 is non empty;
    A7: H2 = rng(F|H1) by A4, Def5;
    the_Vertices_of H1 = F_V"(F_V.:the_Vertices_of H1) by A1, A5, FUNCT_1:94
      .= F_V"rng(F_V|the_Vertices_of H1) by RELAT_1:115
      .= F_V"the_Vertices_of rng(F|H1) by A6, GLIB_010:54
      .= F_V"the_Vertices_of G2 by A3, A7, GLIB_000:def 33
      .= dom F_V by RELSET_1:22
      .= the_Vertices_of G1 by A1, PARTFUN1:def 2;
    then H1 is spanning by GLIB_000:def 33;
    then A8: x in dom f by A4, Th60, RELAT_1:57;
    then H2 = f.x by A4, FUNCT_1:47;
    hence y in rng f by A3, A8, FUNCT_1:3;
  end;
  then G2.allSpanningSG() c= rng f by TARSKI:def 3;
  hence thesis by A2, XBOOLE_0:def 10;
end;
