
theorem Th14:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is onto holds
    (G1 is vertex-finite implies G2 is vertex-finite) &
    (G1 is edge-finite implies G2 is edge-finite)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is onto;
  hereby
    assume A2: G1 is vertex-finite;
    dom F_V is finite by A2;
    then rng F_V is finite by FINSET_1:8;
    hence G2 is vertex-finite by A1, GLIB_010:def 12;
  end;
  assume G1 is edge-finite;
  then dom F_E is finite;
  then rng F_E is finite by FINSET_1:8;
  hence G2 is edge-finite by A1, GLIB_010:def 12;
end;
