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Theorem cmetcvg 19109
Description: The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)
Hypothesis
Ref Expression
iscmet.1  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
cmetcvg  |-  ( ( D  e.  ( CMet `  X )  /\  F  e.  (CauFil `  D )
)  ->  ( J  fLim  F )  =/=  (/) )

Proof of Theorem cmetcvg
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 iscmet.1 . . . 4  |-  J  =  ( MetOpen `  D )
21iscmet 19108 . . 3  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
32simprbi 451 . 2  |-  ( D  e.  ( CMet `  X
)  ->  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) )
4 oveq2 6028 . . . 4  |-  ( f  =  F  ->  ( J  fLim  f )  =  ( J  fLim  F
) )
54neeq1d 2563 . . 3  |-  ( f  =  F  ->  (
( J  fLim  f
)  =/=  (/)  <->  ( J  fLim  F )  =/=  (/) ) )
65rspccva 2994 . 2  |-  ( ( A. f  e.  (CauFil `  D ) ( J 
fLim  f )  =/=  (/)  /\  F  e.  (CauFil `  D ) )  -> 
( J  fLim  F
)  =/=  (/) )
73, 6sylan 458 1  |-  ( ( D  e.  ( CMet `  X )  /\  F  e.  (CauFil `  D )
)  ->  ( J  fLim  F )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   (/)c0 3571   ` cfv 5394  (class class class)co 6020   Metcme 16613   MetOpencmopn 16617    fLim cflim 17887  CauFilccfil 19076   CMetcms 19078
This theorem is referenced by:  cmetcaulem  19112  cmetss  19138  cmetcusp  19175  minveclem4a  19198  fmcncfil  24121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-cmet 19081
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