Theorem cfilufg 20669

Description: The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)

Assertion

Ref	Expression
cfilufg	|- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> ( X filGen F ) e. (CauFilu ` U ) )

Proof of Theorem cfilufg

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

Step	Hyp	Ref	Expression
1		cfilufbas 20665	. . 3 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> F e. ( fBas ` X ) )
2		fgcl 20252	. . 3 |- ( F e. ( fBas ` X ) -> ( X filGen F ) e. ( Fil ` X ) )
3		filfbas 20222	. . 3 |- ( ( X filGen F ) e. ( Fil ` X ) -> ( X filGen F ) e. ( fBas ` X ) )
4	1, 2, 3	3syl 20	. 2 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> ( X filGen F ) e. ( fBas ` X ) )
5	1	ad3antrrr 729	. . . . . . 7 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) /\ v e. U ) /\ b e. F ) /\ ( b X. b ) C_ v ) -> F e. ( fBas ` X ) )
6		ssfg 20246	. . . . . . 7 |- ( F e. ( fBas ` X ) -> F C_ ( X filGen F ) )
7	5, 6	syl 16	. . . . . 6 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) /\ v e. U ) /\ b e. F ) /\ ( b X. b ) C_ v ) -> F C_ ( X filGen F ) )
8		simplr 755	. . . . . 6 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) /\ v e. U ) /\ b e. F ) /\ ( b X. b ) C_ v ) -> b e. F )
9	7, 8	sseldd 3490	. . . . 5 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) /\ v e. U ) /\ b e. F ) /\ ( b X. b ) C_ v ) -> b e. ( X filGen F ) )
10		id 22	. . . . . . . 8 |- ( a = b -> a = b )
11	10	sqxpeqd 5015	. . . . . . 7 |- ( a = b -> ( a X. a ) = ( b X. b ) )
12	11	sseq1d 3516	. . . . . 6 |- ( a = b -> ( ( a X. a ) C_ v <-> ( b X. b ) C_ v ) )
13	12	rspcev 3196	. . . . 5 |- ( ( b e. ( X filGen F ) /\ ( b X. b ) C_ v ) -> E. a e. ( X filGen F ) ( a X. a ) C_ v )
14	9, 13	sylancom 667	. . . 4 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) /\ v e. U ) /\ b e. F ) /\ ( b X. b ) C_ v ) -> E. a e. ( X filGen F ) ( a X. a ) C_ v )
15		iscfilu 20664	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( F e. (CauFilu ` U ) <-> ( F e. ( fBas ` X ) /\ A. v e. U E. b e. F ( b X. b ) C_ v ) ) )
16	15	simplbda 624	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> A. v e. U E. b e. F ( b X. b ) C_ v )
17	16	r19.21bi 2812	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) /\ v e. U ) -> E. b e. F ( b X. b ) C_ v )
18	14, 17	r19.29a 2985	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) /\ v e. U ) -> E. a e. ( X filGen F ) ( a X. a ) C_ v )
19	18	ralrimiva 2857	. 2 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> A. v e. U E. a e. ( X filGen F ) ( a X. a ) C_ v )
20		iscfilu 20664	. . 3 |- ( U e. (UnifOn ` X ) -> ( ( X filGen F ) e. (CauFilu ` U ) <-> ( ( X filGen F ) e. ( fBas ` X ) /\ A. v e. U E. a e. ( X filGen F ) ( a X. a ) C_ v ) ) )
21	20	adantr 465	. 2 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> ( ( X filGen F ) e. (CauFilu ` U ) <-> ( ( X filGen F ) e. ( fBas ` X ) /\ A. v e. U E. a e. ( X filGen F ) ( a X. a ) C_ v ) ) )
22	4, 19, 21	mpbir2and 922	1 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> ( X filGen F ) e. (CauFilu ` U ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1804   A.wral 2793   E.wrex 2794   C_ wss 3461   X. cxp 4987   ` cfv 5578  (class class class)co 6281   fBascfbas 18280   filGencfg 18281   Filcfil 20219  UnifOncust 20575  CauFil_uccfilu 20662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-fbas 18290  df-fg 18291  df-fil 20220  df-ust 20576  df-cfilu 20663

This theorem is referenced by:  ucnextcn  20680