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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.
StepHypRef 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 )
)
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 19561 . . . . . . 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 754 . . . . . 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 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 )
1110, 10xpeq12d 4963 . . . . . . 7  |-  ( a  =  b  ->  (
a  X.  a )  =  ( b  X.  b ) )
1211sseq1d 3481 . . . . . 6  |-  ( a  =  b  ->  (
( a  X.  a
)  C_  v  <->  ( b  X.  b )  C_  v
) )
1312rspcev 3169 . . . . 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 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
) ) )
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 2910 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  /\  v  e.  U
)  ->  E. b  e.  F  ( b  X.  b )  C_  v
)
1814, 17r19.29a 2958 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  /\  v  e.  U
)  ->  E. a  e.  ( X filGen F ) ( a  X.  a
)  C_  v )
1918ralrimiva 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 )
) )
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 913 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 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  CauFiluccfilu 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
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