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Theorem cmetcvg 19199
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 19198 . . 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 6056 . . . 4  |-  ( f  =  F  ->  ( J  fLim  f )  =  ( J  fLim  F
) )
54neeq1d 2588 . . 3  |-  ( f  =  F  ->  (
( J  fLim  f
)  =/=  (/)  <->  ( J  fLim  F )  =/=  (/) ) )
65rspccva 3019 . 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 1721    =/= wne 2575   A.wral 2674   (/)c0 3596   ` cfv 5421  (class class class)co 6048   Metcme 16650   MetOpencmopn 16654    fLim cflim 17927  CauFilccfil 19166   CMetcms 19168
This theorem is referenced by:  cmetcaulem  19202  cmetss  19228  cmetcuspOLD  19268  cmetcusp  19269  minveclem4a  19292  fmcncfil  24278
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-cmet 19171
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