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Theorem cfilfil 21791
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 21789 . 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 621 1  |-  ( ( D  e.  ( *Met `  X )  /\  F  e.  (CauFil `  D ) )  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1826   A.wral 2732   E.wrex 2733    C_ wss 3389    X. cxp 4911   "cima 4916   ` cfv 5496  (class class class)co 6196   0cc0 9403   RR+crp 11139   [,)cico 11452   *Metcxmt 18516   Filcfil 20431  CauFilccfil 21776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  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 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-map 7340  df-xr 9543  df-xmet 18525  df-cfil 21779
This theorem is referenced by:  cfil3i  21793  iscfil3  21797  cfilfcls  21798  iscmet3  21817  cfilresi  21819  cmetss  21838  relcmpcmet  21840  cfilucfil4OLD  21844  cfilucfil4  21845  fmcncfil  28067
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