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Theorem cfilufbas 19879
Description: A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Assertion
Ref Expression
cfilufbas  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  F  e.  (
fBas `  X )
)

Proof of Theorem cfilufbas
Dummy variables  v 
a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscfilu 19878 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  (CauFilu `  U )  <->  ( F  e.  ( fBas `  X
)  /\  A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v
) ) )
21simprbda 623 1  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  F  e.  (
fBas `  X )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2730   E.wrex 2731    C_ wss 3343    X. cxp 4853   ` cfv 5433   fBascfbas 17819  UnifOncust 19789  CauFiluccfilu 19876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-iota 5396  df-fun 5435  df-fn 5436  df-fv 5441  df-ust 19790  df-cfilu 19877
This theorem is referenced by:  fmucnd  19882  cfilufg  19883  cfilucfilOLD  20159  cfilucfil  20160
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