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Theorem cfilufg 18361
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 18357 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  F  e.  (
fBas `  X )
)
2 fgcl 17948 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  e.  ( Fil `  X ) )
3 filfbas 17918 . . 3  |-  ( ( X filGen F )  e.  ( Fil `  X
)  ->  ( X filGen F )  e.  (
fBas `  X )
)
41, 2, 33syl 19 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  ( X filGen F )  e.  ( fBas `  X ) )
51ad3antrrr 712 . . . . . . 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 17942 . . . . . . 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 733 . . . . . 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 3338 . . . . 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 21 . . . . . . . 8  |-  ( a  =  b  ->  a  =  b )
1110, 10xpeq12d 4938 . . . . . . 7  |-  ( a  =  b  ->  (
a  X.  a )  =  ( b  X.  b ) )
1211sseq1d 3364 . . . . . 6  |-  ( a  =  b  ->  (
( a  X.  a
)  C_  v  <->  ( b  X.  b )  C_  v
) )
1312rspcev 3061 . . . . 5  |-  ( ( b  e.  ( X
filGen F )  /\  (
b  X.  b ) 
C_  v )  ->  E. a  e.  ( X filGen F ) ( a  X.  a ) 
C_  v )
149, 13sylancom 650 . . . 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 18356 . . . . . 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 609 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  A. v  e.  U  E. b  e.  F  ( b  X.  b
)  C_  v )
1716r19.21bi 2811 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  /\  v  e.  U
)  ->  E. b  e.  F  ( b  X.  b )  C_  v
)
1814, 17r19.29a 2857 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  /\  v  e.  U
)  ->  E. a  e.  ( X filGen F ) ( a  X.  a
)  C_  v )
1918ralrimiva 2796 . 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 18356 . . 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 453 . 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 890 1  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  ( X filGen F )  e.  (CauFilu `  U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1728   A.wral 2712   E.wrex 2713    C_ wss 3309    X. cxp 4911   ` cfv 5489  (class class class)co 6117   fBascfbas 16727   filGencfg 16728   Filcfil 17915  UnifOncust 18267  CauFiluccfilu 18354
This theorem is referenced by:  ucnextcn  18372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-fbas 16737  df-fg 16738  df-fil 17916  df-ust 18268  df-cfilu 18355
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