Theorem cfilufg 22097

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 22093	. . 3 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> F e. ( fBas ` X ) )
2		fgcl 21682	. . 3 |- ( F e. ( fBas ` X ) -> ( X filGen F ) e. ( Fil ` X ) )
3		filfbas 21652	. . 3 |- ( ( X filGen F ) e. ( Fil ` X ) -> ( X filGen F ) e. ( fBas ` X ) )
4	1, 2, 3	3syl 18	. 2 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> ( X filGen F ) e. ( fBas ` X ) )
5	1	ad3antrrr 766	. . . . . . 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 21676	. . . . . . 7 |- ( F e. ( fBas ` X ) -> F C_ ( X filGen F ) )
7	5, 6	syl 17	. . . . . 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 792	. . . . . 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 3604	. . . . 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 5141	. . . . . . 7 |- ( a = b -> ( a X. a ) = ( b X. b ) )
12	11	sseq1d 3632	. . . . . 6 |- ( a = b -> ( ( a X. a ) C_ v <-> ( b X. b ) C_ v ) )
13	12	rspcev 3309	. . . . 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 701	. . . 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 22092	. . . . . 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 654	. . . . 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 2932	. . . 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 3078	. . 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 2966	. 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 22092	. . 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 481	. 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 957	1 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> ( X filGen F ) e. (CauFilu ` U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   X. cxp 5112   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   filGencfg 19735   Filcfil 21649  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-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-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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744  df-fil 21650  df-ust 22004  df-cfilu 22091

This theorem is referenced by:  ucnextcn  22108