Theorem trcfilu 19891

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 988	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> U e. (UnifOn ` X ) )
2		simp2l 1014	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> F e. (CauFilu ` U ) )
3		iscfilu 19885	. . . . . 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 990	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> A C_ X )
8		simp2r 1015	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> -. (/) e. ( F ↾t A ) )
9		trfbas2 19438	. . . 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 1217	. 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 5739	. . . . . . . . 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 4460	. . . . . . . 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 754	. . . . . . 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 14388	. . . . . . 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 1218	. . . . . 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 4993	. . . . . . 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 3596	. . . . . . . . 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 758	. . . . . . . 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 3414	. . . . . . 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 3422	. . . . . 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, 28	xpeq12d 4886	. . . . . . . 8 |- ( b = ( a i^i A ) -> ( b X. b ) = ( ( a i^i A ) X. ( a i^i A ) ) )
30	29	sseq1d 3404	. . . . . . 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 3094	. . . . . 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 2835	. . . . . . 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 1154	. . . . . 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 1209	. . . . 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 2883	. . . 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 6528	. . . . . 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 14387	. . . . . 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 1217	. . . 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 2883	. . 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 2820	. 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 19826	. . . 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 19885	. . 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 913	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 965   = wceq 1369   e. wcel 1756   A.wral 2736   E.wrex 2737   _Vcvv 2993   i^i cin 3348   C_ wss 3349   (/)c0 3658   X. cxp 4859   ` cfv 5439  (class class class)co 6112   ↾_t crest 14380   fBascfbas 17826  UnifOncust 19796  CauFil_uccfilu 19883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-rest 14382  df-fbas 17836  df-ust 19797  df-cfilu 19884

This theorem is referenced by:  ucnextcn  19901