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Theorem cmetcvg 19238
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 19237 . . 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 6089 . . . 4  |-  ( f  =  F  ->  ( J  fLim  f )  =  ( J  fLim  F
) )
54neeq1d 2614 . . 3  |-  ( f  =  F  ->  (
( J  fLim  f
)  =/=  (/)  <->  ( J  fLim  F )  =/=  (/) ) )
65rspccva 3051 . 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   (/)c0 3628   ` cfv 5454  (class class class)co 6081   Metcme 16687   MetOpencmopn 16691    fLim cflim 17966  CauFilccfil 19205   CMetcms 19207
This theorem is referenced by:  cmetcaulem  19241  cmetss  19267  cmetcuspOLD  19307  cmetcusp  19308  minveclem4a  19331  fmcncfil  24317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-cmet 19210
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