Theorem trcfilu 21358

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 1014	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> U e. (UnifOn ` X ) )
2		simp2l 1040	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> F e. (CauFilu ` U ) )
3		iscfilu 21352	. . . . . 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 491	. . . . 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 671	. . . 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 465	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> F e. ( fBas ` X ) )
7		simp3 1016	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> A C_ X )
8		simp2r 1041	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> -. (/) e. ( F ↾t A ) )
9		trfbas2 20907	. . . 4 |- ( ( F e. ( fBas ` X ) /\ A C_ X ) -> ( ( F ↾t A ) e. ( fBas ` A ) <-> -. (/) e. ( F ↾t A ) ) )
10	9	biimpar 492	. . 3 |- ( ( ( F e. ( fBas ` X ) /\ A C_ X ) /\ -. (/) e. ( F ↾t A ) ) -> ( F ↾t A ) e. ( fBas ` A ) )
11	6, 7, 8, 10	syl21anc 1275	. 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 745	. . . . . . 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 471	. . . . . . . . . 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 5916	. . . . . . . . 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 471	. . . . . . . . 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 4564	. . . . . . . 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 743	. . . . . . 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 767	. . . . . . 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 15376	. . . . . . 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 1276	. . . . . 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 4986	. . . . . . 7 |- ( ( a X. a ) i^i ( A X. A ) ) = ( ( a i^i A ) X. ( a i^i A ) )
22		simpr 467	. . . . . . . . 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 3669	. . . . . . . . 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 17	. . . . . . . 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 774	. . . . . . . 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 3481	. . . . . . 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 3489	. . . . . 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 4879	. . . . . . . 8 |- ( b = ( a i^i A ) -> ( b X. b ) = ( ( a i^i A ) X. ( a i^i A ) ) )
30	29	sseq1d 3471	. . . . . . 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 3162	. . . . . 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 671	. . . . 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 469	. . . . . . . 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 2769	. . . . . . 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 1180	. . . . . 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 1240	. . . . 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 2944	. . . 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 6620	. . . . . 6 |- ( ( A e. _V /\ A e. _V ) -> ( A X. A ) e. _V )
39	16, 16, 38	syl2anc 671	. . . . 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 467	. . . . 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 15375	. . . . . 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 491	. . . . 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 1275	. . . 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 2944	. . 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 2814	. 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 21293	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
47	1, 7, 46	syl2anc 671	. . 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 21352	. . 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 17	. 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 938	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 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   A.wral 2749   E.wrex 2750   _Vcvv 3057   i^i cin 3415   C_ wss 3416   (/)c0 3743   X. cxp 4851   ` cfv 5601  (class class class)co 6315   ↾_t crest 15368   fBascfbas 19007  UnifOncust 21263  CauFil_uccfilu 21350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-rest 15370  df-fbas 19016  df-ust 21264  df-cfilu 21351

This theorem is referenced by:  ucnextcn  21368