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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.
StepHypRef 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 )
)
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 20108 . . . . . . 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 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 )
1110, 10xpeq12d 5024 . . . . . . 7  |-  ( a  =  b  ->  (
a  X.  a )  =  ( b  X.  b ) )
1211sseq1d 3531 . . . . . 6  |-  ( a  =  b  ->  (
( a  X.  a
)  C_  v  <->  ( b  X.  b )  C_  v
) )
1312rspcev 3214 . . . . 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 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
) ) )
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 2833 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  /\  v  e.  U
)  ->  E. b  e.  F  ( b  X.  b )  C_  v
)
1814, 17r19.29a 3003 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  /\  v  e.  U
)  ->  E. a  e.  ( X filGen F ) ( a  X.  a
)  C_  v )
1918ralrimiva 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 )
) )
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 920 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 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  CauFiluccfilu 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
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