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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.
StepHypRef 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 )
)
41, 2, 33syl 20 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  ( X filGen F )  e.  ( fBas `  X ) )
51ad3antrrr 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 ) )
75, 6syl 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 )
97, 8sseldd 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 )
1110sqxpeqd 5015 . . . . . . 7  |-  ( a  =  b  ->  (
a  X.  a )  =  ( b  X.  b ) )
1211sseq1d 3516 . . . . . 6  |-  ( a  =  b  ->  (
( a  X.  a
)  C_  v  <->  ( b  X.  b )  C_  v
) )
1312rspcev 3196 . . . . 5  |-  ( ( b  e.  ( X
filGen F )  /\  (
b  X.  b ) 
C_  v )  ->  E. a  e.  ( X filGen F ) ( a  X.  a ) 
C_  v )
149, 13sylancom 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
) ) )
1615simplbda 624 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  A. v  e.  U  E. b  e.  F  ( b  X.  b
)  C_  v )
1716r19.21bi 2812 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  /\  v  e.  U
)  ->  E. b  e.  F  ( b  X.  b )  C_  v
)
1814, 17r19.29a 2985 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  /\  v  e.  U
)  ->  E. a  e.  ( X filGen F ) ( a  X.  a
)  C_  v )
1918ralrimiva 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 )
) )
2120adantr 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 )
) )
224, 19, 21mpbir2and 922 1  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  ( X filGen F )  e.  (CauFilu `  U
) )
Colors of variables: wff setvar class
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  CauFiluccfilu 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
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