Theorem cfilufg 20531

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 20527	. . 3 |- ( ( U e. (UnifOn ` X ) /\ F e. (CauFilu ` U ) ) -> F e. ( fBas ` X ) )
2		fgcl 20114	. . 3 |- ( F e. ( fBas ` X ) -> ( X filGen F ) e. ( Fil ` X ) )
3		filfbas 20084	. . 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 20108	. . . . . . 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 3505	. . . . 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 5024	. . . . . . 7 |- ( a = b -> ( a X. a ) = ( b X. b ) )
12	11	sseq1d 3531	. . . . . 6 |- ( a = b -> ( ( a X. a ) C_ v <-> ( b X. b ) C_ v ) )
13	12	rspcev 3214	. . . . 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 20526	. . . . . 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 2833	. . . 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 3003	. . 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 2878	. 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 20526	. . 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 920	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 1767   A.wral 2814   E.wrex 2815   C_ wss 3476   X. cxp 4997   ` cfv 5586  (class class class)co 6282   fBascfbas 18177   filGencfg 18178   Filcfil 20081  UnifOncust 20437  CauFil_uccfilu 20524

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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

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-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-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 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18187  df-fg 18188  df-fil 20082  df-ust 20438  df-cfilu 20525

This theorem is referenced by:  ucnextcn  20542