Theorem filnetlem4 32376

Description: Lemma for filnet 32377. (Contributed by Jeff Hankins, 15-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
filnetlem4	|- ( F e. ( Fil ` X ) -> E. d e. DirRel E. f ( f : dom d --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` d ) ) ) )

Distinct variable groups:   x, y   f, d, n, x, y, F   H, d, f, x, y   D, d, f   X, d, f, n

Allowed substitution hints:   D( x, y, n)   H( n)   X( x, y)

Proof of Theorem filnetlem4

Dummy variables k m t v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		filnet.h	. . . . 5 |- H = U_ n e. F ( { n } X. n )
2		filnet.d	. . . . 5 |- D = { <. x , y >. | ( ( x e. H /\ y e. H ) /\ ( 1st ` y ) C_ ( 1st ` x ) ) }
3	1, 2	filnetlem3 32375	. . . 4 |- ( H = U. U. D /\ ( F e. ( Fil ` X ) -> ( H C_ ( F X. X ) /\ D e. DirRel ) ) )
4	3	simpri 478	. . 3 |- ( F e. ( Fil ` X ) -> ( H C_ ( F X. X ) /\ D e. DirRel ) )
5	4	simprd 479	. 2 |- ( F e. ( Fil ` X ) -> D e. DirRel )
6		f2ndres 7191	. . . . 5 |- ( 2nd |` ( F X. X ) ) : ( F X. X ) --> X
7	4	simpld 475	. . . . 5 |- ( F e. ( Fil ` X ) -> H C_ ( F X. X ) )
8		fssres2 6072	. . . . 5 |- ( ( ( 2nd |` ( F X. X ) ) : ( F X. X ) --> X /\ H C_ ( F X. X ) ) -> ( 2nd |` H ) : H --> X )
9	6, 7, 8	sylancr 695	. . . 4 |- ( F e. ( Fil ` X ) -> ( 2nd |` H ) : H --> X )
10		filtop 21659	. . . . . 6 |- ( F e. ( Fil ` X ) -> X e. F )
11		xpexg 6960	. . . . . 6 |- ( ( F e. ( Fil ` X ) /\ X e. F ) -> ( F X. X ) e. _V )
12	10, 11	mpdan 702	. . . . 5 |- ( F e. ( Fil ` X ) -> ( F X. X ) e. _V )
13	12, 7	ssexd 4805	. . . 4 |- ( F e. ( Fil ` X ) -> H e. _V )
14		fex 6490	. . . 4 |- ( ( ( 2nd |` H ) : H --> X /\ H e. _V ) -> ( 2nd |` H ) e. _V )
15	9, 13, 14	syl2anc 693	. . 3 |- ( F e. ( Fil ` X ) -> ( 2nd |` H ) e. _V )
16		dirdm 17234	. . . . . . . 8 |- ( D e. DirRel -> dom D = U. U. D )
17	5, 16	syl 17	. . . . . . 7 |- ( F e. ( Fil ` X ) -> dom D = U. U. D )
18	3	simpli 474	. . . . . . 7 |- H = U. U. D
19	17, 18	syl6reqr 2675	. . . . . 6 |- ( F e. ( Fil ` X ) -> H = dom D )
20	19	feq2d 6031	. . . . 5 |- ( F e. ( Fil ` X ) -> ( ( 2nd |` H ) : H --> X <-> ( 2nd |` H ) : dom D --> X ) )
21	9, 20	mpbid 222	. . . 4 |- ( F e. ( Fil ` X ) -> ( 2nd |` H ) : dom D --> X )
22		eqid 2622	. . . . . . . . . . . . . 14 |- dom D = dom D
23	22	tailf 32370	. . . . . . . . . . . . 13 |- ( D e. DirRel -> ( tail ` D ) : dom D --> ~P dom D )
24	5, 23	syl 17	. . . . . . . . . . . 12 |- ( F e. ( Fil ` X ) -> ( tail ` D ) : dom D --> ~P dom D )
25	19	feq2d 6031	. . . . . . . . . . . 12 |- ( F e. ( Fil ` X ) -> ( ( tail ` D ) : H --> ~P dom D <-> ( tail ` D ) : dom D --> ~P dom D ) )
26	24, 25	mpbird 247	. . . . . . . . . . 11 |- ( F e. ( Fil ` X ) -> ( tail ` D ) : H --> ~P dom D )
27	26	adantr 481	. . . . . . . . . 10 |- ( ( F e. ( Fil ` X ) /\ t C_ X ) -> ( tail ` D ) : H --> ~P dom D )
28		ffn 6045	. . . . . . . . . 10 |- ( ( tail ` D ) : H --> ~P dom D -> ( tail ` D ) Fn H )
29		imaeq2 5462	. . . . . . . . . . . 12 |- ( d = ( ( tail ` D ) ` f ) -> ( ( 2nd |` H ) " d ) = ( ( 2nd |` H ) " ( ( tail ` D ) ` f ) ) )
30	29	sseq1d 3632	. . . . . . . . . . 11 |- ( d = ( ( tail ` D ) ` f ) -> ( ( ( 2nd |` H ) " d ) C_ t <-> ( ( 2nd |` H ) " ( ( tail ` D ) ` f ) ) C_ t ) )
31	30	rexrn 6361	. . . . . . . . . 10 |- ( ( tail ` D ) Fn H -> ( E. d e. ran ( tail ` D ) ( ( 2nd |` H ) " d ) C_ t <-> E. f e. H ( ( 2nd |` H ) " ( ( tail ` D ) ` f ) ) C_ t ) )
32	27, 28, 31	3syl 18	. . . . . . . . 9 |- ( ( F e. ( Fil ` X ) /\ t C_ X ) -> ( E. d e. ran ( tail ` D ) ( ( 2nd |` H ) " d ) C_ t <-> E. f e. H ( ( 2nd |` H ) " ( ( tail ` D ) ` f ) ) C_ t ) )
33		fo2nd 7189	. . . . . . . . . . . . . . 15 |- 2nd : _V -onto-> _V
34		fofn 6117	. . . . . . . . . . . . . . 15 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
35	33, 34	ax-mp 5	. . . . . . . . . . . . . 14 |- 2nd Fn _V
36		ssv 3625	. . . . . . . . . . . . . 14 |- H C_ _V
37		fnssres 6004	. . . . . . . . . . . . . 14 |- ( ( 2nd Fn _V /\ H C_ _V ) -> ( 2nd |` H ) Fn H )
38	35, 36, 37	mp2an 708	. . . . . . . . . . . . 13 |- ( 2nd |` H ) Fn H
39		fnfun 5988	. . . . . . . . . . . . 13 |- ( ( 2nd |` H ) Fn H -> Fun ( 2nd |` H ) )
40	38, 39	ax-mp 5	. . . . . . . . . . . 12 |- Fun ( 2nd |` H )
41	27	ffvelrnda 6359	. . . . . . . . . . . . . . 15 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( tail ` D ) ` f ) e. ~P dom D )
42	41	elpwid 4170	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( tail ` D ) ` f ) C_ dom D )
43	19	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> H = dom D )
44	42, 43	sseqtr4d 3642	. . . . . . . . . . . . 13 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( tail ` D ) ` f ) C_ H )
45		fndm 5990	. . . . . . . . . . . . . 14 |- ( ( 2nd |` H ) Fn H -> dom ( 2nd |` H ) = H )
46	38, 45	ax-mp 5	. . . . . . . . . . . . 13 |- dom ( 2nd |` H ) = H
47	44, 46	syl6sseqr 3652	. . . . . . . . . . . 12 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( tail ` D ) ` f ) C_ dom ( 2nd |` H ) )
48		funimass4 6247	. . . . . . . . . . . 12 |- ( ( Fun ( 2nd |` H ) /\ ( ( tail ` D ) ` f ) C_ dom ( 2nd |` H ) ) -> ( ( ( 2nd |` H ) " ( ( tail ` D ) ` f ) ) C_ t <-> A. d e. ( ( tail ` D ) ` f ) ( ( 2nd |` H ) ` d ) e. t ) )
49	40, 47, 48	sylancr 695	. . . . . . . . . . 11 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( ( 2nd |` H ) " ( ( tail ` D ) ` f ) ) C_ t <-> A. d e. ( ( tail ` D ) ` f ) ( ( 2nd |` H ) ` d ) e. t ) )
50	5	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> D e. DirRel )
51		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> f e. H )
52	51, 43	eleqtrd 2703	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> f e. dom D )
53		vex 3203	. . . . . . . . . . . . . . . . 17 |- d e. _V
54	53	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> d e. _V )
55	22	eltail 32369	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ f e. dom D /\ d e. _V ) -> ( d e. ( ( tail ` D ) ` f ) <-> f D d ) )
56	50, 52, 54, 55	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( d e. ( ( tail ` D ) ` f ) <-> f D d ) )
57	51	biantrurd 529	. . . . . . . . . . . . . . . . 17 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( d e. H <-> ( f e. H /\ d e. H ) ) )
58	57	anbi1d 741	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) <-> ( ( f e. H /\ d e. H ) /\ ( 1st ` d ) C_ ( 1st ` f ) ) ) )
59		vex 3203	. . . . . . . . . . . . . . . . 17 |- f e. _V
60	1, 2, 59, 53	filnetlem1 32373	. . . . . . . . . . . . . . . 16 |- ( f D d <-> ( ( f e. H /\ d e. H ) /\ ( 1st ` d ) C_ ( 1st ` f ) ) )
61	58, 60	syl6bbr 278	. . . . . . . . . . . . . . 15 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) <-> f D d ) )
62	56, 61	bitr4d 271	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( d e. ( ( tail ` D ) ` f ) <-> ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) ) )
63	62	imbi1d 331	. . . . . . . . . . . . 13 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( d e. ( ( tail ` D ) ` f ) -> ( ( 2nd |` H ) ` d ) e. t ) <-> ( ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) -> ( ( 2nd |` H ) ` d ) e. t ) ) )
64		fvres 6207	. . . . . . . . . . . . . . . . 17 |- ( d e. H -> ( ( 2nd |` H ) ` d ) = ( 2nd ` d ) )
65	64	eleq1d 2686	. . . . . . . . . . . . . . . 16 |- ( d e. H -> ( ( ( 2nd |` H ) ` d ) e. t <-> ( 2nd ` d ) e. t ) )
66	65	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) -> ( ( ( 2nd |` H ) ` d ) e. t <-> ( 2nd ` d ) e. t ) )
67	66	pm5.74i 260	. . . . . . . . . . . . . 14 |- ( ( ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) -> ( ( 2nd |` H ) ` d ) e. t ) <-> ( ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) -> ( 2nd ` d ) e. t ) )
68		impexp 462	. . . . . . . . . . . . . 14 |- ( ( ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) -> ( 2nd ` d ) e. t ) <-> ( d e. H -> ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) ) )
69	67, 68	bitri 264	. . . . . . . . . . . . 13 |- ( ( ( d e. H /\ ( 1st ` d ) C_ ( 1st ` f ) ) -> ( ( 2nd |` H ) ` d ) e. t ) <-> ( d e. H -> ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) ) )
70	63, 69	syl6bb 276	. . . . . . . . . . . 12 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( d e. ( ( tail ` D ) ` f ) -> ( ( 2nd |` H ) ` d ) e. t ) <-> ( d e. H -> ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) ) ) )
71	70	ralbidv2 2984	. . . . . . . . . . 11 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( A. d e. ( ( tail ` D ) ` f ) ( ( 2nd |` H ) ` d ) e. t <-> A. d e. H ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) ) )
72	49, 71	bitrd 268	. . . . . . . . . 10 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ f e. H ) -> ( ( ( 2nd |` H ) " ( ( tail ` D ) ` f ) ) C_ t <-> A. d e. H ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) ) )
73	72	rexbidva 3049	. . . . . . . . 9 |- ( ( F e. ( Fil ` X ) /\ t C_ X ) -> ( E. f e. H ( ( 2nd |` H ) " ( ( tail ` D ) ` f ) ) C_ t <-> E. f e. H A. d e. H ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) ) )
74		vex 3203	. . . . . . . . . . . . . . . . 17 |- k e. _V
75		vex 3203	. . . . . . . . . . . . . . . . 17 |- v e. _V
76	74, 75	op1std 7178	. . . . . . . . . . . . . . . 16 |- ( d = <. k , v >. -> ( 1st ` d ) = k )
77	76	sseq1d 3632	. . . . . . . . . . . . . . 15 |- ( d = <. k , v >. -> ( ( 1st ` d ) C_ ( 1st ` f ) <-> k C_ ( 1st ` f ) ) )
78	74, 75	op2ndd 7179	. . . . . . . . . . . . . . . 16 |- ( d = <. k , v >. -> ( 2nd ` d ) = v )
79	78	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( d = <. k , v >. -> ( ( 2nd ` d ) e. t <-> v e. t ) )
80	77, 79	imbi12d 334	. . . . . . . . . . . . . 14 |- ( d = <. k , v >. -> ( ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) <-> ( k C_ ( 1st ` f ) -> v e. t ) ) )
81	80	raliunxp 5261	. . . . . . . . . . . . 13 |- ( A. d e. U_ k e. F ( { k } X. k ) ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) <-> A. k e. F A. v e. k ( k C_ ( 1st ` f ) -> v e. t ) )
82		sneq 4187	. . . . . . . . . . . . . . . . 17 |- ( n = k -> { n } = { k } )
83		id 22	. . . . . . . . . . . . . . . . 17 |- ( n = k -> n = k )
84	82, 83	xpeq12d 5140	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( { n } X. n ) = ( { k } X. k ) )
85	84	cbviunv 4559	. . . . . . . . . . . . . . 15 |- U_ n e. F ( { n } X. n ) = U_ k e. F ( { k } X. k )
86	1, 85	eqtri 2644	. . . . . . . . . . . . . 14 |- H = U_ k e. F ( { k } X. k )
87	86	raleqi 3142	. . . . . . . . . . . . 13 |- ( A. d e. H ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) <-> A. d e. U_ k e. F ( { k } X. k ) ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) )
88		dfss3 3592	. . . . . . . . . . . . . . . 16 |- ( k C_ t <-> A. v e. k v e. t )
89	88	imbi2i 326	. . . . . . . . . . . . . . 15 |- ( ( k C_ ( 1st ` f ) -> k C_ t ) <-> ( k C_ ( 1st ` f ) -> A. v e. k v e. t ) )
90		r19.21v 2960	. . . . . . . . . . . . . . 15 |- ( A. v e. k ( k C_ ( 1st ` f ) -> v e. t ) <-> ( k C_ ( 1st ` f ) -> A. v e. k v e. t ) )
91	89, 90	bitr4i 267	. . . . . . . . . . . . . 14 |- ( ( k C_ ( 1st ` f ) -> k C_ t ) <-> A. v e. k ( k C_ ( 1st ` f ) -> v e. t ) )
92	91	ralbii 2980	. . . . . . . . . . . . 13 |- ( A. k e. F ( k C_ ( 1st ` f ) -> k C_ t ) <-> A. k e. F A. v e. k ( k C_ ( 1st ` f ) -> v e. t ) )
93	81, 87, 92	3bitr4i 292	. . . . . . . . . . . 12 |- ( A. d e. H ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) <-> A. k e. F ( k C_ ( 1st ` f ) -> k C_ t ) )
94	93	rexbii 3041	. . . . . . . . . . 11 |- ( E. f e. H A. d e. H ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) <-> E. f e. H A. k e. F ( k C_ ( 1st ` f ) -> k C_ t ) )
95	1	rexeqi 3143	. . . . . . . . . . 11 |- ( E. f e. H A. k e. F ( k C_ ( 1st ` f ) -> k C_ t ) <-> E. f e. U_ n e. F ( { n } X. n ) A. k e. F ( k C_ ( 1st ` f ) -> k C_ t ) )
96		vex 3203	. . . . . . . . . . . . . . . 16 |- n e. _V
97		vex 3203	. . . . . . . . . . . . . . . 16 |- m e. _V
98	96, 97	op1std 7178	. . . . . . . . . . . . . . 15 |- ( f = <. n , m >. -> ( 1st ` f ) = n )
99	98	sseq2d 3633	. . . . . . . . . . . . . 14 |- ( f = <. n , m >. -> ( k C_ ( 1st ` f ) <-> k C_ n ) )
100	99	imbi1d 331	. . . . . . . . . . . . 13 |- ( f = <. n , m >. -> ( ( k C_ ( 1st ` f ) -> k C_ t ) <-> ( k C_ n -> k C_ t ) ) )
101	100	ralbidv 2986	. . . . . . . . . . . 12 |- ( f = <. n , m >. -> ( A. k e. F ( k C_ ( 1st ` f ) -> k C_ t ) <-> A. k e. F ( k C_ n -> k C_ t ) ) )
102	101	rexiunxp 5262	. . . . . . . . . . 11 |- ( E. f e. U_ n e. F ( { n } X. n ) A. k e. F ( k C_ ( 1st ` f ) -> k C_ t ) <-> E. n e. F E. m e. n A. k e. F ( k C_ n -> k C_ t ) )
103	94, 95, 102	3bitri 286	. . . . . . . . . 10 |- ( E. f e. H A. d e. H ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) <-> E. n e. F E. m e. n A. k e. F ( k C_ n -> k C_ t ) )
104		fileln0 21654	. . . . . . . . . . . . . 14 |- ( ( F e. ( Fil ` X ) /\ n e. F ) -> n =/= (/) )
105	104	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ n e. F ) -> n =/= (/) )
106		r19.9rzv 4065	. . . . . . . . . . . . 13 |- ( n =/= (/) -> ( A. k e. F ( k C_ n -> k C_ t ) <-> E. m e. n A. k e. F ( k C_ n -> k C_ t ) ) )
107	105, 106	syl 17	. . . . . . . . . . . 12 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ n e. F ) -> ( A. k e. F ( k C_ n -> k C_ t ) <-> E. m e. n A. k e. F ( k C_ n -> k C_ t ) ) )
108		ssid 3624	. . . . . . . . . . . . . . 15 |- n C_ n
109		sseq1 3626	. . . . . . . . . . . . . . . . 17 |- ( k = n -> ( k C_ n <-> n C_ n ) )
110		sseq1 3626	. . . . . . . . . . . . . . . . 17 |- ( k = n -> ( k C_ t <-> n C_ t ) )
111	109, 110	imbi12d 334	. . . . . . . . . . . . . . . 16 |- ( k = n -> ( ( k C_ n -> k C_ t ) <-> ( n C_ n -> n C_ t ) ) )
112	111	rspcv 3305	. . . . . . . . . . . . . . 15 |- ( n e. F -> ( A. k e. F ( k C_ n -> k C_ t ) -> ( n C_ n -> n C_ t ) ) )
113	108, 112	mpii 46	. . . . . . . . . . . . . 14 |- ( n e. F -> ( A. k e. F ( k C_ n -> k C_ t ) -> n C_ t ) )
114	113	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ n e. F ) -> ( A. k e. F ( k C_ n -> k C_ t ) -> n C_ t ) )
115		sstr2 3610	. . . . . . . . . . . . . . 15 |- ( k C_ n -> ( n C_ t -> k C_ t ) )
116	115	com12 32	. . . . . . . . . . . . . 14 |- ( n C_ t -> ( k C_ n -> k C_ t ) )
117	116	ralrimivw 2967	. . . . . . . . . . . . 13 |- ( n C_ t -> A. k e. F ( k C_ n -> k C_ t ) )
118	114, 117	impbid1 215	. . . . . . . . . . . 12 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ n e. F ) -> ( A. k e. F ( k C_ n -> k C_ t ) <-> n C_ t ) )
119	107, 118	bitr3d 270	. . . . . . . . . . 11 |- ( ( ( F e. ( Fil ` X ) /\ t C_ X ) /\ n e. F ) -> ( E. m e. n A. k e. F ( k C_ n -> k C_ t ) <-> n C_ t ) )
120	119	rexbidva 3049	. . . . . . . . . 10 |- ( ( F e. ( Fil ` X ) /\ t C_ X ) -> ( E. n e. F E. m e. n A. k e. F ( k C_ n -> k C_ t ) <-> E. n e. F n C_ t ) )
121	103, 120	syl5bb 272	. . . . . . . . 9 |- ( ( F e. ( Fil ` X ) /\ t C_ X ) -> ( E. f e. H A. d e. H ( ( 1st ` d ) C_ ( 1st ` f ) -> ( 2nd ` d ) e. t ) <-> E. n e. F n C_ t ) )
122	32, 73, 121	3bitrd 294	. . . . . . . 8 |- ( ( F e. ( Fil ` X ) /\ t C_ X ) -> ( E. d e. ran ( tail ` D ) ( ( 2nd |` H ) " d ) C_ t <-> E. n e. F n C_ t ) )
123	122	pm5.32da 673	. . . . . . 7 |- ( F e. ( Fil ` X ) -> ( ( t C_ X /\ E. d e. ran ( tail ` D ) ( ( 2nd |` H ) " d ) C_ t ) <-> ( t C_ X /\ E. n e. F n C_ t ) ) )
124		filn0 21666	. . . . . . . . . . . 12 |- ( F e. ( Fil ` X ) -> F =/= (/) )
125	96	snnz 4309	. . . . . . . . . . . . . . . 16 |- { n } =/= (/)
126	104, 125	jctil 560	. . . . . . . . . . . . . . 15 |- ( ( F e. ( Fil ` X ) /\ n e. F ) -> ( { n } =/= (/) /\ n =/= (/) ) )
127		neanior 2886	. . . . . . . . . . . . . . 15 |- ( ( { n } =/= (/) /\ n =/= (/) ) <-> -. ( { n } = (/) \/ n = (/) ) )
128	126, 127	sylib 208	. . . . . . . . . . . . . 14 |- ( ( F e. ( Fil ` X ) /\ n e. F ) -> -. ( { n } = (/) \/ n = (/) ) )
129		ss0b 3973	. . . . . . . . . . . . . . 15 |- ( ( { n } X. n ) C_ (/) <-> ( { n } X. n ) = (/) )
130		xpeq0 5554	. . . . . . . . . . . . . . 15 |- ( ( { n } X. n ) = (/) <-> ( { n } = (/) \/ n = (/) ) )
131	129, 130	bitri 264	. . . . . . . . . . . . . 14 |- ( ( { n } X. n ) C_ (/) <-> ( { n } = (/) \/ n = (/) ) )
132	128, 131	sylnibr 319	. . . . . . . . . . . . 13 |- ( ( F e. ( Fil ` X ) /\ n e. F ) -> -. ( { n } X. n ) C_ (/) )
133	132	ralrimiva 2966	. . . . . . . . . . . 12 |- ( F e. ( Fil ` X ) -> A. n e. F -. ( { n } X. n ) C_ (/) )
134		r19.2z 4060	. . . . . . . . . . . 12 |- ( ( F =/= (/) /\ A. n e. F -. ( { n } X. n ) C_ (/) ) -> E. n e. F -. ( { n } X. n ) C_ (/) )
135	124, 133, 134	syl2anc 693	. . . . . . . . . . 11 |- ( F e. ( Fil ` X ) -> E. n e. F -. ( { n } X. n ) C_ (/) )
136		rexnal 2995	. . . . . . . . . . 11 |- ( E. n e. F -. ( { n } X. n ) C_ (/) <-> -. A. n e. F ( { n } X. n ) C_ (/) )
137	135, 136	sylib 208	. . . . . . . . . 10 |- ( F e. ( Fil ` X ) -> -. A. n e. F ( { n } X. n ) C_ (/) )
138	1	sseq1i 3629	. . . . . . . . . . . 12 |- ( H C_ (/) <-> U_ n e. F ( { n } X. n ) C_ (/) )
139		ss0b 3973	. . . . . . . . . . . 12 |- ( H C_ (/) <-> H = (/) )
140		iunss 4561	. . . . . . . . . . . 12 |- ( U_ n e. F ( { n } X. n ) C_ (/) <-> A. n e. F ( { n } X. n ) C_ (/) )
141	138, 139, 140	3bitr3i 290	. . . . . . . . . . 11 |- ( H = (/) <-> A. n e. F ( { n } X. n ) C_ (/) )
142	141	necon3abii 2840	. . . . . . . . . 10 |- ( H =/= (/) <-> -. A. n e. F ( { n } X. n ) C_ (/) )
143	137, 142	sylibr 224	. . . . . . . . 9 |- ( F e. ( Fil ` X ) -> H =/= (/) )
144		dmresi 5457	. . . . . . . . . . . 12 |- dom ( _I |` H ) = H
145	1, 2	filnetlem2 32374	. . . . . . . . . . . . . 14 |- ( ( _I |` H ) C_ D /\ D C_ ( H X. H ) )
146	145	simpli 474	. . . . . . . . . . . . 13 |- ( _I |` H ) C_ D
147		dmss 5323	. . . . . . . . . . . . 13 |- ( ( _I |` H ) C_ D -> dom ( _I |` H ) C_ dom D )
148	146, 147	ax-mp 5	. . . . . . . . . . . 12 |- dom ( _I |` H ) C_ dom D
149	144, 148	eqsstr3i 3636	. . . . . . . . . . 11 |- H C_ dom D
150	145	simpri 478	. . . . . . . . . . . . 13 |- D C_ ( H X. H )
151		dmss 5323	. . . . . . . . . . . . 13 |- ( D C_ ( H X. H ) -> dom D C_ dom ( H X. H ) )
152	150, 151	ax-mp 5	. . . . . . . . . . . 12 |- dom D C_ dom ( H X. H )
153		dmxpid 5345	. . . . . . . . . . . 12 |- dom ( H X. H ) = H
154	152, 153	sseqtri 3637	. . . . . . . . . . 11 |- dom D C_ H
155	149, 154	eqssi 3619	. . . . . . . . . 10 |- H = dom D
156	155	tailfb 32372	. . . . . . . . 9 |- ( ( D e. DirRel /\ H =/= (/) ) -> ran ( tail ` D ) e. ( fBas ` H ) )
157	5, 143, 156	syl2anc 693	. . . . . . . 8 |- ( F e. ( Fil ` X ) -> ran ( tail ` D ) e. ( fBas ` H ) )
158		elfm 21751	. . . . . . . 8 |- ( ( X e. F /\ ran ( tail ` D ) e. ( fBas ` H ) /\ ( 2nd |` H ) : H --> X ) -> ( t e. ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) <-> ( t C_ X /\ E. d e. ran ( tail ` D ) ( ( 2nd |` H ) " d ) C_ t ) ) )
159	10, 157, 9, 158	syl3anc 1326	. . . . . . 7 |- ( F e. ( Fil ` X ) -> ( t e. ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) <-> ( t C_ X /\ E. d e. ran ( tail ` D ) ( ( 2nd |` H ) " d ) C_ t ) ) )
160		filfbas 21652	. . . . . . . 8 |- ( F e. ( Fil ` X ) -> F e. ( fBas ` X ) )
161		elfg 21675	. . . . . . . 8 |- ( F e. ( fBas ` X ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. n e. F n C_ t ) ) )
162	160, 161	syl 17	. . . . . . 7 |- ( F e. ( Fil ` X ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. n e. F n C_ t ) ) )
163	123, 159, 162	3bitr4d 300	. . . . . 6 |- ( F e. ( Fil ` X ) -> ( t e. ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) <-> t e. ( X filGen F ) ) )
164	163	eqrdv 2620	. . . . 5 |- ( F e. ( Fil ` X ) -> ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) = ( X filGen F ) )
165		fgfil 21679	. . . . 5 |- ( F e. ( Fil ` X ) -> ( X filGen F ) = F )
166	164, 165	eqtr2d 2657	. . . 4 |- ( F e. ( Fil ` X ) -> F = ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) )
167	21, 166	jca 554	. . 3 |- ( F e. ( Fil ` X ) -> ( ( 2nd |` H ) : dom D --> X /\ F = ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) ) )
168		feq1 6026	. . . . 5 |- ( f = ( 2nd |` H ) -> ( f : dom D --> X <-> ( 2nd |` H ) : dom D --> X ) )
169		oveq2 6658	. . . . . . 7 |- ( f = ( 2nd |` H ) -> ( X FilMap f ) = ( X FilMap ( 2nd |` H ) ) )
170	169	fveq1d 6193	. . . . . 6 |- ( f = ( 2nd |` H ) -> ( ( X FilMap f ) ` ran ( tail ` D ) ) = ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) )
171	170	eqeq2d 2632	. . . . 5 |- ( f = ( 2nd |` H ) -> ( F = ( ( X FilMap f ) ` ran ( tail ` D ) ) <-> F = ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) ) )
172	168, 171	anbi12d 747	. . . 4 |- ( f = ( 2nd |` H ) -> ( ( f : dom D --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` D ) ) ) <-> ( ( 2nd |` H ) : dom D --> X /\ F = ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) ) ) )
173	172	spcegv 3294	. . 3 |- ( ( 2nd |` H ) e. _V -> ( ( ( 2nd |` H ) : dom D --> X /\ F = ( ( X FilMap ( 2nd |` H ) ) ` ran ( tail ` D ) ) ) -> E. f ( f : dom D --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` D ) ) ) ) )
174	15, 167, 173	sylc 65	. 2 |- ( F e. ( Fil ` X ) -> E. f ( f : dom D --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` D ) ) ) )
175		dmeq 5324	. . . . . 6 |- ( d = D -> dom d = dom D )
176	175	feq2d 6031	. . . . 5 |- ( d = D -> ( f : dom d --> X <-> f : dom D --> X ) )
177		fveq2 6191	. . . . . . . 8 |- ( d = D -> ( tail ` d ) = ( tail ` D ) )
178	177	rneqd 5353	. . . . . . 7 |- ( d = D -> ran ( tail ` d ) = ran ( tail ` D ) )
179	178	fveq2d 6195	. . . . . 6 |- ( d = D -> ( ( X FilMap f ) ` ran ( tail ` d ) ) = ( ( X FilMap f ) ` ran ( tail ` D ) ) )
180	179	eqeq2d 2632	. . . . 5 |- ( d = D -> ( F = ( ( X FilMap f ) ` ran ( tail ` d ) ) <-> F = ( ( X FilMap f ) ` ran ( tail ` D ) ) ) )
181	176, 180	anbi12d 747	. . . 4 |- ( d = D -> ( ( f : dom d --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` d ) ) ) <-> ( f : dom D --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` D ) ) ) ) )
182	181	exbidv 1850	. . 3 |- ( d = D -> ( E. f ( f : dom d --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` d ) ) ) <-> E. f ( f : dom D --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` D ) ) ) ) )
183	182	rspcev 3309	. 2 |- ( ( D e. DirRel /\ E. f ( f : dom D --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` D ) ) ) ) -> E. d e. DirRel E. f ( f : dom d --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` d ) ) ) )
184	5, 174, 183	syl2anc 693	1 |- ( F e. ( Fil ` X ) -> E. d e. DirRel E. f ( f : dom d --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` d ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   {copab 4712   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   DirRelcdir 17228   tailctail 17229   fBascfbas 19734   filGencfg 19735   Filcfil 21649   FilMap cfm 21737

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-rep 4771  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-dir 17230  df-tail 17231  df-fbas 19743  df-fg 19744  df-fil 21650  df-fm 21742

This theorem is referenced by:  filnet  32377