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Theorem iscmet 21849
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 5899 . 2  |-  ( D  e.  ( CMet `  X
)  ->  X  e.  _V )
2 elfvex 5899 . . 3  |-  ( D  e.  ( Met `  X
)  ->  X  e.  _V )
32adantr 465 . 2  |-  ( ( D  e.  ( Met `  X )  /\  A. f  e.  (CauFil `  D
) ( J  fLim  f )  =/=  (/) )  ->  X  e.  _V )
4 fveq2 5872 . . . . . 6  |-  ( x  =  X  ->  ( Met `  x )  =  ( Met `  X
) )
5 rabeq 3103 . . . . . 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 21822 . . . . 5  |-  CMet  =  ( x  e.  _V  |->  { d  e.  ( Met `  x )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
8 fvex 5882 . . . . . 6  |-  ( Met `  X )  e.  _V
98rabex 4607 . . . . 5  |-  { d  e.  ( Met `  X
)  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) }  e.  _V
106, 7, 9fvmpt 5956 . . . 4  |-  ( X  e.  _V  ->  ( CMet `  X )  =  { d  e.  ( Met `  X )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
1110eleq2d 2527 . . 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 5872 . . . . 5  |-  ( d  =  D  ->  (CauFil `  d )  =  (CauFil `  D ) )
13 fveq2 5872 . . . . . . . 8  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  ( MetOpen `  D )
)
14 iscmet.1 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
1513, 14syl6eqr 2516 . . . . . . 7  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  J )
1615oveq1d 6311 . . . . . 6  |-  ( d  =  D  ->  (
( MetOpen `  d )  fLim  f )  =  ( J  fLim  f )
)
1716neeq1d 2734 . . . . 5  |-  ( d  =  D  ->  (
( ( MetOpen `  d
)  fLim  f )  =/=  (/)  <->  ( J  fLim  f )  =/=  (/) ) )
1812, 17raleqbidv 3068 . . . 4  |-  ( d  =  D  ->  ( A. f  e.  (CauFil `  d ) ( (
MetOpen `  d )  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
1918elrab 3257 . . 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 261 . 2  |-  ( X  e.  _V  ->  ( D  e.  ( CMet `  X )  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) ) )
211, 3, 20pm5.21nii 353 1  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109   (/)c0 3793   ` cfv 5594  (class class class)co 6296   Metcme 18531   MetOpencmopn 18535    fLim cflim 20561  CauFilccfil 21817   CMetcms 21819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-cmet 21822
This theorem is referenced by:  cmetcvg  21850  cmetmet  21851  iscmet3  21858  cmetss  21879  equivcmet  21880  relcmpcmet  21881  cmetcusp1OLD  21917  cmetcusp1  21918
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