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Theorem cfilfil 20896
Description: A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
cfilfil  |-  ( ( D  e.  ( *Met `  X )  /\  F  e.  (CauFil `  D ) )  ->  F  e.  ( Fil `  X ) )

Proof of Theorem cfilfil
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscfil 20894 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( F  e.  (CauFil `  D
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  F  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
21simprbda 623 1  |-  ( ( D  e.  ( *Met `  X )  /\  F  e.  (CauFil `  D ) )  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   A.wral 2795   E.wrex 2796    C_ wss 3428    X. cxp 4938   "cima 4943   ` cfv 5518  (class class class)co 6192   0cc0 9385   RR+crp 11094   [,)cico 11405   *Metcxmt 17912   Filcfil 19536  CauFilccfil 20881
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 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442
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-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-map 7318  df-xr 9525  df-xmet 17921  df-cfil 20884
This theorem is referenced by:  cfil3i  20898  iscfil3  20902  cfilfcls  20903  iscmet3  20922  cfilresi  20924  cmetss  20943  relcmpcmet  20945  cfilucfil4OLD  20949  cfilucfil4  20950  fmcncfil  26497
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