
theorem Th103:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2, V being set
  holds F_E.:G1.edgesBetween(V) c= G2.edgesBetween(F_V.:V)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2, V be set;
  now
    let e be object;
    assume e in F_E.:G1.edgesBetween(V);
    then consider e0 being object such that
      A1: e0 in dom F_E & e0 in G1.edgesBetween(V) & F_E.e0 = e
      by FUNCT_1:def 6;
    set v0 = (the_Source_of G1).e0, w0 = (the_Target_of G1).e0;
    A2: v0 in dom F_V & w0 in dom F_V by A1, GLIB_010:5;
    e0 Joins v0,w0,G1 by A1, GLIB_000:def 13;
    then A3: e Joins F_V.v0,F_V.w0,G2 by A1, A2, GLIB_010:4;
    v0 in V & w0 in V by A1, GLIB_000:31;
    then F_V.v0 in F_V.:V & F_V.w0 in F_V.:V by A2, FUNCT_1:def 6;
    hence e in G2.edgesBetween(F_V.:V) by A3, GLIB_000:32;
  end;
  hence thesis by TARSKI:def 3;
end;
