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Theorem filnetlem3 32375
Description: Lemma for filnet 32377. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypotheses
Ref Expression
filnet.h |- H = U_ n e. F ( { n } X. n )
filnet.d |- D = { <. x , y >. | ( ( x e. H /\ y e. H ) /\ ( 1st ` y ) C_ ( 1st ` x ) ) }
Assertion
Ref Expression
filnetlem3 |- ( H = U. U. D /\ ( F e. ( Fil ` X ) -> ( H C_ ( F X. X ) /\ D e. DirRel ) ) )
Distinct variable groups:   x, y, n, F   x, H, y   n, X
Allowed substitution hints:   D( x, y, n)   H( n)   X( x, y)
Proof of Theorem filnetlem3
Dummy variables u v w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmresi 5457 . . . . . 6 |- dom ( _I |` H ) = H
2 filnet.h . . . . . . . . 9 |- H = U_ n e. F ( { n } X. n )
3 filnet.d . . . . . . . . 9 |- D = { <. x , y >. | ( ( x e. H /\ y e. H ) /\ ( 1st ` y ) C_ ( 1st ` x ) ) }
42, 3filnetlem2 32374 . . . . . . . 8 |- ( ( _I |` H ) C_ D /\ D C_ ( H X. H ) )
54simpli 474 . . . . . . 7 |- ( _I |` H ) C_ D
6 dmss 5323 . . . . . . 7 |- ( ( _I |` H ) C_ D -> dom ( _I |` H ) C_ dom D )
75, 6ax-mp 5 . . . . . 6 |- dom ( _I |` H ) C_ dom D
81, 7eqsstr3i 3636 . . . . 5 |- H C_ dom D
9 ssun1 3776 . . . . 5 |- dom D C_ ( dom D u. ran D )
108, 9sstri 3612 . . . 4 |- H C_ ( dom D u. ran D )
11 dmrnssfld 5384 . . . 4 |- ( dom D u. ran D ) C_ U. U. D
1210, 11sstri 3612 . . 3 |- H C_ U. U. D
134simpri 478 . . . . 5 |- D C_ ( H X. H )
14 uniss 4458 . . . . 5 |- ( D C_ ( H X. H ) -> U. D C_ U. ( H X. H ) )
15 uniss 4458 . . . . 5 |- ( U. D C_ U. ( H X. H ) -> U. U. D C_ U. U. ( H X. H ) )
1613, 14, 15mp2b 10 . . . 4 |- U. U. D C_ U. U. ( H X. H )
17 unixpss 5234 . . . . 5 |- U. U. ( H X. H ) C_ ( H u. H )
18 unidm 3756 . . . . 5 |- ( H u. H ) = H
1917, 18sseqtri 3637 . . . 4 |- U. U. ( H X. H ) C_ H
2016, 19sstri 3612 . . 3 |- U. U. D C_ H
2112, 20eqssi 3619 . 2 |- H = U. U. D
22 filelss 21656 . . . . . . . 8 |- ( ( F e. ( Fil ` X ) /\ n e. F ) -> n C_ X )
23 xpss2 5229 . . . . . . . 8 |- ( n C_ X -> ( { n } X. n ) C_ ( { n } X. X ) )
2422, 23syl 17 . . . . . . 7 |- ( ( F e. ( Fil ` X ) /\ n e. F ) -> ( { n } X. n ) C_ ( { n } X. X ) )
2524ralrimiva 2966 . . . . . 6 |- ( F e. ( Fil ` X ) -> A. n e. F ( { n } X. n ) C_ ( { n } X. X ) )
26 ss2iun 4536 . . . . . 6 |- ( A. n e. F ( { n } X. n ) C_ ( { n } X. X ) -> U_ n e. F ( { n } X. n ) C_ U_ n e. F ( { n } X. X ) )
2725, 26syl 17 . . . . 5 |- ( F e. ( Fil ` X ) -> U_ n e. F ( { n } X. n ) C_ U_ n e. F ( { n } X. X ) )
28 iunxpconst 5175 . . . . 5 |- U_ n e. F ( { n } X. X ) = ( F X. X )
2927, 28syl6sseq 3651 . . . 4 |- ( F e. ( Fil ` X ) -> U_ n e. F ( { n } X. n ) C_ ( F X. X ) )
302, 29syl5eqss 3649 . . 3 |- ( F e. ( Fil ` X ) -> H C_ ( F X. X ) )
315a1i 11 . . . . 5 |- ( F e. ( Fil ` X ) -> ( _I |` H ) C_ D )
323relopabi 5245 . . . . 5 |- Rel D
3331, 32jctil 560 . . . 4 |- ( F e. ( Fil ` X ) -> ( Rel D /\ ( _I |` H ) C_ D ) )
34 simpl 473 . . . . . . . . . 10 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> F e. ( Fil ` X ) )
3530adantr 481 . . . . . . . . . . . 12 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> H C_ ( F X. X ) )
36 simprl 794 . . . . . . . . . . . 12 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> v e. H )
3735, 36sseldd 3604 . . . . . . . . . . 11 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> v e. ( F X. X ) )
38 xp1st 7198 . . . . . . . . . . 11 |- ( v e. ( F X. X ) -> ( 1st ` v ) e. F )
3937, 38syl 17 . . . . . . . . . 10 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> ( 1st ` v ) e. F )
40 simprr 796 . . . . . . . . . . . 12 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> z e. H )
4135, 40sseldd 3604 . . . . . . . . . . 11 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> z e. ( F X. X ) )
42 xp1st 7198 . . . . . . . . . . 11 |- ( z e. ( F X. X ) -> ( 1st ` z ) e. F )
4341, 42syl 17 . . . . . . . . . 10 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> ( 1st ` z ) e. F )
44 filinn0 21664 . . . . . . . . . 10 |- ( ( F e. ( Fil ` X ) /\ ( 1st ` v ) e. F /\ ( 1st ` z ) e. F ) -> ( ( 1st ` v ) i^i ( 1st ` z ) ) =/= (/) )
4534, 39, 43, 44syl3anc 1326 . . . . . . . . 9 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> ( ( 1st ` v ) i^i ( 1st ` z ) ) =/= (/) )
46 n0 3931 . . . . . . . . 9 |- ( ( ( 1st ` v ) i^i ( 1st ` z ) ) =/= (/) <-> E. u u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) )
4745, 46sylib 208 . . . . . . . 8 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> E. u u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) )
4836adantr 481 . . . . . . . . . 10 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> v e. H )
49 filin 21658 . . . . . . . . . . . . . 14 |- ( ( F e. ( Fil ` X ) /\ ( 1st ` v ) e. F /\ ( 1st ` z ) e. F ) -> ( ( 1st ` v ) i^i ( 1st ` z ) ) e. F )
5034, 39, 43, 49syl3anc 1326 . . . . . . . . . . . . 13 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> ( ( 1st ` v ) i^i ( 1st ` z ) ) e. F )
5150adantr 481 . . . . . . . . . . . 12 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> ( ( 1st ` v ) i^i ( 1st ` z ) ) e. F )
52 simpr 477 . . . . . . . . . . . 12 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) )
53 id 22 . . . . . . . . . . . . 13 |- ( n = ( ( 1st ` v ) i^i ( 1st ` z ) ) -> n = ( ( 1st ` v ) i^i ( 1st ` z ) ) )
5453opeliunxp2 5260 . . . . . . . . . . . 12 |- ( <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. e. U_ n e. F ( { n } X. n ) <-> ( ( ( 1st ` v ) i^i ( 1st ` z ) ) e. F /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) )
5551, 52, 54sylanbrc 698 . . . . . . . . . . 11 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. e. U_ n e. F ( { n } X. n ) )
5655, 2syl6eleqr 2712 . . . . . . . . . 10 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. e. H )
57 fvex 6201 . . . . . . . . . . . . . 14 |- ( 1st ` v ) e. _V
5857inex1 4799 . . . . . . . . . . . . 13 |- ( ( 1st ` v ) i^i ( 1st ` z ) ) e. _V
59 vex 3203 . . . . . . . . . . . . 13 |- u e. _V
6058, 59op1st 7176 . . . . . . . . . . . 12 |- ( 1st ` <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) = ( ( 1st ` v ) i^i ( 1st ` z ) )
61 inss1 3833 . . . . . . . . . . . 12 |- ( ( 1st ` v ) i^i ( 1st ` z ) ) C_ ( 1st ` v )
6260, 61eqsstri 3635 . . . . . . . . . . 11 |- ( 1st ` <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) C_ ( 1st ` v )
63 vex 3203 . . . . . . . . . . . 12 |- v e. _V
64 opex 4932 . . . . . . . . . . . 12 |- <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. e. _V
652, 3, 63, 64filnetlem1 32373 . . . . . . . . . . 11 |- ( v D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. <-> ( ( v e. H /\ <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. e. H ) /\ ( 1st ` <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) C_ ( 1st ` v ) ) )
6662, 65mpbiran2 954 . . . . . . . . . 10 |- ( v D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. <-> ( v e. H /\ <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. e. H ) )
6748, 56, 66sylanbrc 698 . . . . . . . . 9 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> v D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. )
6840adantr 481 . . . . . . . . . 10 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> z e. H )
69 inss2 3834 . . . . . . . . . . . 12 |- ( ( 1st ` v ) i^i ( 1st ` z ) ) C_ ( 1st ` z )
7060, 69eqsstri 3635 . . . . . . . . . . 11 |- ( 1st ` <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) C_ ( 1st ` z )
71 vex 3203 . . . . . . . . . . . 12 |- z e. _V
722, 3, 71, 64filnetlem1 32373 . . . . . . . . . . 11 |- ( z D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. <-> ( ( z e. H /\ <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. e. H ) /\ ( 1st ` <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) C_ ( 1st ` z ) ) )
7370, 72mpbiran2 954 . . . . . . . . . 10 |- ( z D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. <-> ( z e. H /\ <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. e. H ) )
7468, 56, 73sylanbrc 698 . . . . . . . . 9 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> z D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. )
75 breq2 4657 . . . . . . . . . . 11 |- ( w = <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. -> ( v D w <-> v D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) )
76 breq2 4657 . . . . . . . . . . 11 |- ( w = <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. -> ( z D w <-> z D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) )
7775, 76anbi12d 747 . . . . . . . . . 10 |- ( w = <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. -> ( ( v D w /\ z D w ) <-> ( v D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. /\ z D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) ) )
7864, 77spcev 3300 . . . . . . . . 9 |- ( ( v D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. /\ z D <. ( ( 1st ` v ) i^i ( 1st ` z ) ) , u >. ) -> E. w ( v D w /\ z D w ) )
7967, 74, 78syl2anc 693 . . . . . . . 8 |- ( ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) /\ u e. ( ( 1st ` v ) i^i ( 1st ` z ) ) ) -> E. w ( v D w /\ z D w ) )
8047, 79exlimddv 1863 . . . . . . 7 |- ( ( F e. ( Fil ` X ) /\ ( v e. H /\ z e. H ) ) -> E. w ( v D w /\ z D w ) )
8180ralrimivva 2971 . . . . . 6 |- ( F e. ( Fil ` X ) -> A. v e. H A. z e. H E. w ( v D w /\ z D w ) )
82 codir 5516 . . . . . 6 |- ( ( H X. H ) C_ ( `' D o. D ) <-> A. v e. H A. z e. H E. w ( v D w /\ z D w ) )
8381, 82sylibr 224 . . . . 5 |- ( F e. ( Fil ` X ) -> ( H X. H ) C_ ( `' D o. D ) )
84 vex 3203 . . . . . . . . . . . . 13 |- w e. _V
852, 3, 63, 84filnetlem1 32373 . . . . . . . . . . . 12 |- ( v D w <-> ( ( v e. H /\ w e. H ) /\ ( 1st ` w ) C_ ( 1st ` v ) ) )
8685simplbi 476 . . . . . . . . . . 11 |- ( v D w -> ( v e. H /\ w e. H ) )
8786simpld 475 . . . . . . . . . 10 |- ( v D w -> v e. H )
882, 3, 84, 71filnetlem1 32373 . . . . . . . . . . . 12 |- ( w D z <-> ( ( w e. H /\ z e. H ) /\ ( 1st ` z ) C_ ( 1st ` w ) ) )
8988simplbi 476 . . . . . . . . . . 11 |- ( w D z -> ( w e. H /\ z e. H ) )
9089simprd 479 . . . . . . . . . 10 |- ( w D z -> z e. H )
9187, 90anim12i 590 . . . . . . . . 9 |- ( ( v D w /\ w D z ) -> ( v e. H /\ z e. H ) )
9288simprbi 480 . . . . . . . . . 10 |- ( w D z -> ( 1st ` z ) C_ ( 1st ` w ) )
9385simprbi 480 . . . . . . . . . 10 |- ( v D w -> ( 1st ` w ) C_ ( 1st ` v ) )
9492, 93sylan9ssr 3617 . . . . . . . . 9 |- ( ( v D w /\ w D z ) -> ( 1st ` z ) C_ ( 1st ` v ) )
952, 3, 63, 71filnetlem1 32373 . . . . . . . . 9 |- ( v D z <-> ( ( v e. H /\ z e. H ) /\ ( 1st ` z ) C_ ( 1st ` v ) ) )
9691, 94, 95sylanbrc 698 . . . . . . . 8 |- ( ( v D w /\ w D z ) -> v D z )
9796ax-gen 1722 . . . . . . 7 |- A. z ( ( v D w /\ w D z ) -> v D z )
9897gen2 1723 . . . . . 6 |- A. v A. w A. z ( ( v D w /\ w D z ) -> v D z )
99 cotr 5508 . . . . . 6 |- ( ( D o. D ) C_ D <-> A. v A. w A. z ( ( v D w /\ w D z ) -> v D z ) )
10098, 99mpbir 221 . . . . 5 |- ( D o. D ) C_ D
10183, 100jctil 560 . . . 4 |- ( F e. ( Fil ` X ) -> ( ( D o. D ) C_ D /\ ( H X. H ) C_ ( `' D o. D ) ) )
102 filtop 21659 . . . . . . . . 9 |- ( F e. ( Fil ` X ) -> X e. F )
103 xpexg 6960 . . . . . . . . 9 |- ( ( F e. ( Fil ` X ) /\ X e. F ) -> ( F X. X ) e. _V )
104102, 103mpdan 702 . . . . . . . 8 |- ( F e. ( Fil ` X ) -> ( F X. X ) e. _V )
105104, 30ssexd 4805 . . . . . . 7 |- ( F e. ( Fil ` X ) -> H e. _V )
106 xpexg 6960 . . . . . . 7 |- ( ( H e. _V /\ H e. _V ) -> ( H X. H ) e. _V )
107105, 105, 106syl2anc 693 . . . . . 6 |- ( F e. ( Fil ` X ) -> ( H X. H ) e. _V )
108 ssexg 4804 . . . . . 6 |- ( ( D C_ ( H X. H ) /\ ( H X. H ) e. _V ) -> D e. _V )
10913, 107, 108sylancr 695 . . . . 5 |- ( F e. ( Fil ` X ) -> D e. _V )
11021isdir 17232 . . . . 5 |- ( D e. _V -> ( D e. DirRel <-> ( ( Rel D /\ ( _I |` H ) C_ D ) /\ ( ( D o. D ) C_ D /\ ( H X. H ) C_ ( `' D o. D ) ) ) ) )
111109, 110syl 17 . . . 4 |- ( F e. ( Fil ` X ) -> ( D e. DirRel <-> ( ( Rel D /\ ( _I |` H ) C_ D ) /\ ( ( D o. D ) C_ D /\ ( H X. H ) C_ ( `' D o. D ) ) ) ) )
11233, 101, 111mpbir2and 957 . . 3 |- ( F e. ( Fil ` X ) -> D e. DirRel )
11330, 112jca 554 . 2 |- ( F e. ( Fil ` X ) -> ( H C_ ( F X. X ) /\ D e. DirRel ) )
11421, 113pm3.2i 471 1 |- ( H = U. U. D /\ ( F e. ( Fil ` X ) -> ( H C_ ( F X. X ) /\ D e. DirRel ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   {copab 4712   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Rel wrel 5119   ` cfv 5888   1stc1st 7166   DirRelcdir 17228   Filcfil 21649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-dir 17230  df-fbas 19743  df-fil 21650
This theorem is referenced by:  filnetlem4  32376
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