
theorem Th87:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v being Vertex of G1 st F is directed & v in dom F_V holds
    F_E.:(v.edgesIn()) c= (F_V/.v).edgesIn() &
    F_E.:(v.edgesOut()) c= (F_V/.v).edgesOut()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let v be Vertex of G1;
  assume A1: F is directed & v in dom F_V;
  now
    let e be object;
    assume e in F_E.:(v.edgesIn());
    then consider e0 being object such that
      A2: e0 in dom F_E & e0 in v.edgesIn() & e = F_E.e0 by FUNCT_1:def 6;
    A3: (the_Target_of G1).e0 = v by A2, GLIB_000:56;
    set w = (the_Source_of G1).e0;
    A4: w in dom F_V by A2, GLIB_010:5;
    e0 DJoins w,v,G1 by A2, A3, GLIB_000:def 14;
    then F_E.e0 DJoins F_V.w,F_V.v,G2 by A1, A2, A4, GLIB_010:def 14;
    then F_E.e0 DJoins F_V.w,F_V/.v,G2 by A1, PARTFUN1:def 6;
    hence e in (F_V/.v).edgesIn() by A2, GLIB_000:57;
  end;
  hence F_E.:(v.edgesIn()) c= (F_V/.v).edgesIn() by TARSKI:def 3;
  now
    let e be object;
    assume e in F_E.:(v.edgesOut());
    then consider e0 being object such that
      A5: e0 in dom F_E & e0 in v.edgesOut() & e = F_E.e0 by FUNCT_1:def 6;
    A6: (the_Source_of G1).e0 = v by A5, GLIB_000:58;
    set w = (the_Target_of G1).e0;
    A7: w in dom F_V by A5, GLIB_010:5;
    e0 DJoins v,w,G1 by A5, A6, GLIB_000:def 14;
    then F_E.e0 DJoins F_V.v,F_V.w,G2 by A1, A5, A7, GLIB_010:def 14;
    then F_E.e0 DJoins F_V/.v,F_V.w,G2 by A1, PARTFUN1:def 6;
    hence e in (F_V/.v).edgesOut() by A5, GLIB_000:59;
  end;
  hence F_E.:(v.edgesOut()) c= (F_V/.v).edgesOut() by TARSKI:def 3;
end;
