Theorem trcfilu 20922

Description: Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)

Assertion

Ref	Expression
trcfilu	|- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> ( F ↾t A ) e. (CauFilu ` ( U ↾t ( A X. A ) ) ) )

Proof of Theorem trcfilu

Dummy variables a b v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 996	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> U e. (UnifOn ` X ) )
2		simp2l 1022	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> F e. (CauFilu ` U ) )
3		iscfilu 20916	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( F e. (CauFilu ` U ) <-> ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) ) )
4	3	biimpa 484	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) )
5	1, 2, 4	syl2anc 661	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) )
6	5	simpld 459	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> F e. ( fBas ` X ) )
7		simp3 998	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> A C_ X )
8		simp2r 1023	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> -. (/) e. ( F ↾t A ) )
9		trfbas2 20469	. . . 4 |- ( ( F e. ( fBas ` X ) /\ A C_ X ) -> ( ( F ↾t A ) e. ( fBas ` A ) <-> -. (/) e. ( F ↾t A ) ) )
10	9	biimpar 485	. . 3 |- ( ( ( F e. ( fBas ` X ) /\ A C_ X ) /\ -. (/) e. ( F ↾t A ) ) -> ( F ↾t A ) e. ( fBas ` A ) )
11	6, 7, 8, 10	syl21anc 1227	. 2 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> ( F ↾t A ) e. ( fBas ` A ) )
12	2	ad5antr 733	. . . . . . 7 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> F e. (CauFilu ` U ) )
13	1	adantr 465	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) -> U e. (UnifOn ` X ) )
14	13	elfvexd 5900	. . . . . . . . 9 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) -> X e. _V )
15	7	adantr 465	. . . . . . . . 9 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) -> A C_ X )
16	14, 15	ssexd 4603	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) -> A e. _V )
17	16	ad4antr 731	. . . . . . 7 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> A e. _V )
18		simplr 755	. . . . . . 7 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> a e. F )
19		elrestr 14845	. . . . . . 7 |- ( ( F e. (CauFilu ` U ) /\ A e. _V /\ a e. F ) -> ( a i^i A ) e. ( F ↾t A ) )
20	12, 17, 18, 19	syl3anc 1228	. . . . . 6 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> ( a i^i A ) e. ( F ↾t A ) )
21		inxp 5145	. . . . . . 7 |- ( ( a X. a ) i^i ( A X. A ) ) = ( ( a i^i A ) X. ( a i^i A ) )
22		simpr 461	. . . . . . . . 9 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> ( a X. a ) C_ v )
23		ssrin 3719	. . . . . . . . 9 |- ( ( a X. a ) C_ v -> ( ( a X. a ) i^i ( A X. A ) ) C_ ( v i^i ( A X. A ) ) )
24	22, 23	syl 16	. . . . . . . 8 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> ( ( a X. a ) i^i ( A X. A ) ) C_ ( v i^i ( A X. A ) ) )
25		simpllr 760	. . . . . . . 8 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> w = ( v i^i ( A X. A ) ) )
26	24, 25	sseqtr4d 3536	. . . . . . 7 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> ( ( a X. a ) i^i ( A X. A ) ) C_ w )
27	21, 26	syl5eqssr 3544	. . . . . 6 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> ( ( a i^i A ) X. ( a i^i A ) ) C_ w )
28		id 22	. . . . . . . . 9 |- ( b = ( a i^i A ) -> b = ( a i^i A ) )
29	28	sqxpeqd 5034	. . . . . . . 8 |- ( b = ( a i^i A ) -> ( b X. b ) = ( ( a i^i A ) X. ( a i^i A ) ) )
30	29	sseq1d 3526	. . . . . . 7 |- ( b = ( a i^i A ) -> ( ( b X. b ) C_ w <-> ( ( a i^i A ) X. ( a i^i A ) ) C_ w ) )
31	30	rspcev 3210	. . . . . 6 |- ( ( ( a i^i A ) e. ( F ↾t A ) /\ ( ( a i^i A ) X. ( a i^i A ) ) C_ w ) -> E. b e. ( F ↾t A ) ( b X. b ) C_ w )
32	20, 27, 31	syl2anc 661	. . . . 5 |- ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) /\ a e. F ) /\ ( a X. a ) C_ v ) -> E. b e. ( F ↾t A ) ( b X. b ) C_ w )
33	5	simprd 463	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> A. v e. U E. a e. F ( a X. a ) C_ v )
34	33	r19.21bi 2826	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ v e. U ) -> E. a e. F ( a X. a ) C_ v )
35	34	3ad2antr2 1162	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ ( w e. ( U ↾t ( A X. A ) ) /\ v e. U /\ w = ( v i^i ( A X. A ) ) ) ) -> E. a e. F ( a X. a ) C_ v )
36	35	3anassrs 1218	. . . . 5 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) -> E. a e. F ( a X. a ) C_ v )
37	32, 36	r19.29a 2999	. . . 4 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) /\ v e. U ) /\ w = ( v i^i ( A X. A ) ) ) -> E. b e. ( F ↾t A ) ( b X. b ) C_ w )
38		xpexg 6601	. . . . . 6 |- ( ( A e. _V /\ A e. _V ) -> ( A X. A ) e. _V )
39	16, 16, 38	syl2anc 661	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) -> ( A X. A ) e. _V )
40		simpr 461	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) -> w e. ( U ↾t ( A X. A ) ) )
41		elrest 14844	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ ( A X. A ) e. _V ) -> ( w e. ( U ↾t ( A X. A ) ) <-> E. v e. U w = ( v i^i ( A X. A ) ) ) )
42	41	biimpa 484	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ ( A X. A ) e. _V ) /\ w e. ( U ↾t ( A X. A ) ) ) -> E. v e. U w = ( v i^i ( A X. A ) ) )
43	13, 39, 40, 42	syl21anc 1227	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) -> E. v e. U w = ( v i^i ( A X. A ) ) )
44	37, 43	r19.29a 2999	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) /\ w e. ( U ↾t ( A X. A ) ) ) -> E. b e. ( F ↾t A ) ( b X. b ) C_ w )
45	44	ralrimiva 2871	. 2 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> A. w e. ( U ↾t ( A X. A ) ) E. b e. ( F ↾t A ) ( b X. b ) C_ w )
46		trust 20857	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
47	1, 7, 46	syl2anc 661	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
48		iscfilu 20916	. . 3 |- ( ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) -> ( ( F ↾t A ) e. (CauFilu ` ( U ↾t ( A X. A ) ) ) <-> ( ( F ↾t A ) e. ( fBas ` A ) /\ A. w e. ( U ↾t ( A X. A ) ) E. b e. ( F ↾t A ) ( b X. b ) C_ w ) ) )
49	47, 48	syl 16	. 2 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> ( ( F ↾t A ) e. (CauFilu ` ( U ↾t ( A X. A ) ) ) <-> ( ( F ↾t A ) e. ( fBas ` A ) /\ A. w e. ( U ↾t ( A X. A ) ) E. b e. ( F ↾t A ) ( b X. b ) C_ w ) ) )
50	11, 45, 49	mpbir2and 922	1 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> ( F ↾t A ) e. (CauFilu ` ( U ↾t ( A X. A ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109   i^i cin 3470   C_ wss 3471   (/)c0 3793   X. cxp 5006   ` cfv 5594  (class class class)co 6296   ↾_t crest 14837   fBascfbas 18532  UnifOncust 20827  CauFil_uccfilu 20914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-rest 14839  df-fbas 18542  df-ust 20828  df-cfilu 20915

This theorem is referenced by:  ucnextcn  20932