Theorem cfilufg 19984

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 19980	. . 3 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> F e. ( fBas ` X ) )
2		fgcl 19567	. . 3 |- ( F e. ( fBas ` X ) -> ( X filGen F ) e. ( Fil ` X ) )
3		filfbas 19537	. . 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 19561	. . . . . . 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 754	. . . . . 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 3455	. . . . 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, 10	xpeq12d 4963	. . . . . . 7 |- ( a = b -> ( a X. a ) = ( b X. b ) )
12	11	sseq1d 3481	. . . . . 6 |- ( a = b -> ( ( a X. a ) C_ v <-> ( b X. b ) C_ v ) )
13	12	rspcev 3169	. . . . 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 19979	. . . . . 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 2910	. . . 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 2958	. . 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 2822	. 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 19979	. . 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 913	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 1758   A.wral 2795   E.wrex 2796   C_ wss 3426   X. cxp 4936   ` cfv 5516  (class class class)co 6190   fBascfbas 17913   filGencfg 17914   Filcfil 19534  UnifOncust 19890  CauFil_uccfilu 19977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-fbas 17923  df-fg 17924  df-fil 19535  df-ust 19891  df-cfilu 19978

This theorem is referenced by:  ucnextcn  19995