Theorem trcfilu 20624

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 20618	. . . . . 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 20171	. . . 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 5894	. . . . . . . . 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 4594	. . . . . . . 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 14687	. . . . . . 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 5135	. . . . . . 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 3723	. . . . . . . . 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 3541	. . . . . . 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 3549	. . . . . 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 5024	. . . . . . . 8 |- ( b = ( a i^i A ) -> ( b X. b ) = ( ( a i^i A ) X. ( a i^i A ) ) )
30	29	sseq1d 3531	. . . . . . 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 3214	. . . . . 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 2833	. . . . . . 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 3003	. . . 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 6587	. . . . . 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 14686	. . . . . 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 3003	. . 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 2878	. 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 20559	. . . 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 20618	. . 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 920	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 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   (/)c0 3785   X. cxp 4997   ` cfv 5588  (class class class)co 6285   ↾_t crest 14679   fBascfbas 18217  UnifOncust 20529  CauFil_uccfilu 20616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-rest 14681  df-fbas 18227  df-ust 20530  df-cfilu 20617

This theorem is referenced by:  ucnextcn  20634