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Theorem iscmet 19229
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 5750 . 2  |-  ( D  e.  ( CMet `  X
)  ->  X  e.  _V )
2 elfvex 5750 . . 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 5720 . . . . . 6  |-  ( x  =  X  ->  ( Met `  x )  =  ( Met `  X
) )
5 rabeq 2942 . . . . . 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 19202 . . . . 5  |-  CMet  =  ( x  e.  _V  |->  { d  e.  ( Met `  x )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
8 fvex 5734 . . . . . 6  |-  ( Met `  X )  e.  _V
98rabex 4346 . . . . 5  |-  { d  e.  ( Met `  X
)  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) }  e.  _V
106, 7, 9fvmpt 5798 . . . 4  |-  ( X  e.  _V  ->  ( CMet `  X )  =  { d  e.  ( Met `  X )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
1110eleq2d 2502 . . 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 5720 . . . . 5  |-  ( d  =  D  ->  (CauFil `  d )  =  (CauFil `  D ) )
13 fveq2 5720 . . . . . . . 8  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  ( MetOpen `  D )
)
14 iscmet.1 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
1513, 14syl6eqr 2485 . . . . . . 7  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  J )
1615oveq1d 6088 . . . . . 6  |-  ( d  =  D  ->  (
( MetOpen `  d )  fLim  f )  =  ( J  fLim  f )
)
1716neeq1d 2611 . . . . 5  |-  ( d  =  D  ->  (
( ( MetOpen `  d
)  fLim  f )  =/=  (/)  <->  ( J  fLim  f )  =/=  (/) ) )
1812, 17raleqbidv 2908 . . . 4  |-  ( d  =  D  ->  ( A. f  e.  (CauFil `  d ) ( (
MetOpen `  d )  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
1918elrab 3084 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701   _Vcvv 2948   (/)c0 3620   ` cfv 5446  (class class class)co 6073   Metcme 16679   MetOpencmopn 16683    fLim cflim 17958  CauFilccfil 19197   CMetcms 19199
This theorem is referenced by:  cmetcvg  19230  cmetmet  19231  iscmet3  19238  cmetss  19259  equivcmet  19260  relcmpcmet  19261  cmetcusp1OLD  19297  cmetcusp1  19298
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-cmet 19202
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