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Theorem cmetcvg 18727
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 18726 . . 3  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
32simprbi 450 . 2  |-  ( D  e.  ( CMet `  X
)  ->  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) )
4 oveq2 5882 . . . 4  |-  ( f  =  F  ->  ( J  fLim  f )  =  ( J  fLim  F
) )
54neeq1d 2472 . . 3  |-  ( f  =  F  ->  (
( J  fLim  f
)  =/=  (/)  <->  ( J  fLim  F )  =/=  (/) ) )
65rspccva 2896 . 2  |-  ( ( A. f  e.  (CauFil `  D ) ( J 
fLim  f )  =/=  (/)  /\  F  e.  (CauFil `  D ) )  -> 
( J  fLim  F
)  =/=  (/) )
73, 6sylan 457 1  |-  ( ( D  e.  ( CMet `  X )  /\  F  e.  (CauFil `  D )
)  ->  ( J  fLim  F )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   (/)c0 3468   ` cfv 5271  (class class class)co 5874   Metcme 16386   MetOpencmopn 16388    fLim cflim 17645  CauFilccfil 18694   CMetcms 18696
This theorem is referenced by:  cmetcaulem  18730  cmetss  18756  minveclem4a  18810
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-cmet 18699
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