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Theorem iscmet 19108
Description: The property " D is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypothesis
Ref Expression
iscmet.1  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
iscmet  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
Distinct variable groups:    D, f    f, J    f, X

Proof of Theorem iscmet
Dummy variables  d  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5698 . 2  |-  ( D  e.  ( CMet `  X
)  ->  X  e.  _V )
2 elfvex 5698 . . 3  |-  ( D  e.  ( Met `  X
)  ->  X  e.  _V )
32adantr 452 . 2  |-  ( ( D  e.  ( Met `  X )  /\  A. f  e.  (CauFil `  D
) ( J  fLim  f )  =/=  (/) )  ->  X  e.  _V )
4 fveq2 5668 . . . . . 6  |-  ( x  =  X  ->  ( Met `  x )  =  ( Met `  X
) )
5 rabeq 2893 . . . . . 6  |-  ( ( Met `  x )  =  ( Met `  X
)  ->  { d  e.  ( Met `  x
)  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) }  =  {
d  e.  ( Met `  X )  |  A. f  e.  (CauFil `  d
) ( ( MetOpen `  d )  fLim  f
)  =/=  (/) } )
64, 5syl 16 . . . . 5  |-  ( x  =  X  ->  { d  e.  ( Met `  x
)  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) }  =  {
d  e.  ( Met `  X )  |  A. f  e.  (CauFil `  d
) ( ( MetOpen `  d )  fLim  f
)  =/=  (/) } )
7 df-cmet 19081 . . . . 5  |-  CMet  =  ( x  e.  _V  |->  { d  e.  ( Met `  x )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
8 fvex 5682 . . . . . 6  |-  ( Met `  X )  e.  _V
98rabex 4295 . . . . 5  |-  { d  e.  ( Met `  X
)  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) }  e.  _V
106, 7, 9fvmpt 5745 . . . 4  |-  ( X  e.  _V  ->  ( CMet `  X )  =  { d  e.  ( Met `  X )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
1110eleq2d 2454 . . 3  |-  ( X  e.  _V  ->  ( D  e.  ( CMet `  X )  <->  D  e.  { d  e.  ( Met `  X )  |  A. f  e.  (CauFil `  d
) ( ( MetOpen `  d )  fLim  f
)  =/=  (/) } ) )
12 fveq2 5668 . . . . 5  |-  ( d  =  D  ->  (CauFil `  d )  =  (CauFil `  D ) )
13 fveq2 5668 . . . . . . . 8  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  ( MetOpen `  D )
)
14 iscmet.1 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
1513, 14syl6eqr 2437 . . . . . . 7  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  J )
1615oveq1d 6035 . . . . . 6  |-  ( d  =  D  ->  (
( MetOpen `  d )  fLim  f )  =  ( J  fLim  f )
)
1716neeq1d 2563 . . . . 5  |-  ( d  =  D  ->  (
( ( MetOpen `  d
)  fLim  f )  =/=  (/)  <->  ( J  fLim  f )  =/=  (/) ) )
1812, 17raleqbidv 2859 . . . 4  |-  ( d  =  D  ->  ( A. f  e.  (CauFil `  d ) ( (
MetOpen `  d )  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
1918elrab 3035 . . 3  |-  ( D  e.  { d  e.  ( Met `  X
)  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) }  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
2011, 19syl6bb 253 . 2  |-  ( X  e.  _V  ->  ( D  e.  ( CMet `  X )  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) ) )
211, 3, 20pm5.21nii 343 1  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   {crab 2653   _Vcvv 2899   (/)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:  cmetcvg  19109  cmetmet  19110  iscmet3  19117  cmetss  19138  equivcmet  19139  relcmpcmet  19140  cmetcusp1  19174
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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