Theorem trcfilu 22098

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 1061	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> U e. (UnifOn ` X ) )
2		simp2l 1087	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> F e. (CauFilu ` U ) )
3		iscfilu 22092	. . . . . 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 501	. . . . 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 693	. . . 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 475	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> F e. ( fBas ` X ) )
7		simp3 1063	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> A C_ X )
8		simp2r 1088	. . 3 |- ( ( U e. (UnifOn ` X ) /\ ( F e. (CauFilu ` U ) /\ -. (/) e. ( F ↾t A ) ) /\ A C_ X ) -> -. (/) e. ( F ↾t A ) )
9		trfbas2 21647	. . . 4 |- ( ( F e. ( fBas ` X ) /\ A C_ X ) -> ( ( F ↾t A ) e. ( fBas ` A ) <-> -. (/) e. ( F ↾t A ) ) )
10	9	biimpar 502	. . 3 |- ( ( ( F e. ( fBas ` X ) /\ A C_ X ) /\ -. (/) e. ( F ↾t A ) ) -> ( F ↾t A ) e. ( fBas ` A ) )
11	6, 7, 8, 10	syl21anc 1325	. 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 770	. . . . . . 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 481	. . . . . . . . . 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 6222	. . . . . . . . 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 481	. . . . . . . . 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 4805	. . . . . . . 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 768	. . . . . . 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 792	. . . . . . 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 16089	. . . . . . 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 1326	. . . . . 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 5254	. . . . . . 7 |- ( ( a X. a ) i^i ( A X. A ) ) = ( ( a i^i A ) X. ( a i^i A ) )
22		simpr 477	. . . . . . . . 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 3838	. . . . . . . . 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 799	. . . . . . . 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 3642	. . . . . . 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 3650	. . . . . 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 5141	. . . . . . . 8 |- ( b = ( a i^i A ) -> ( b X. b ) = ( ( a i^i A ) X. ( a i^i A ) ) )
30	29	sseq1d 3632	. . . . . . 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 3309	. . . . . 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 693	. . . . 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 479	. . . . . . . 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 2932	. . . . . . 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 1227	. . . . . 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 1290	. . . . 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 3078	. . . 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 6960	. . . . . 6 |- ( ( A e. _V /\ A e. _V ) -> ( A X. A ) e. _V )
39	16, 16, 38	syl2anc 693	. . . . 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 477	. . . . 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 16088	. . . . . 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 501	. . . . 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 1325	. . . 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 3078	. . 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 2966	. 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 22033	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
47	1, 7, 46	syl2anc 693	. . 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 22092	. . 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 957	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   X. cxp 5112   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   fBascfbas 19734  UnifOncust 22003  CauFil_uccfilu 22090

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-rest 16083  df-fbas 19743  df-ust 22004  df-cfilu 22091

This theorem is referenced by:  ucnextcn  22108