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Theorem iscmet 20795
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 5717 . 2  |-  ( D  e.  ( CMet `  X
)  ->  X  e.  _V )
2 elfvex 5717 . . 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 5691 . . . . . 6  |-  ( x  =  X  ->  ( Met `  x )  =  ( Met `  X
) )
5 rabeq 2966 . . . . . 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 20768 . . . . 5  |-  CMet  =  ( x  e.  _V  |->  { d  e.  ( Met `  x )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
8 fvex 5701 . . . . . 6  |-  ( Met `  X )  e.  _V
98rabex 4443 . . . . 5  |-  { d  e.  ( Met `  X
)  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) }  e.  _V
106, 7, 9fvmpt 5774 . . . 4  |-  ( X  e.  _V  ->  ( CMet `  X )  =  { d  e.  ( Met `  X )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
1110eleq2d 2510 . . 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 5691 . . . . 5  |-  ( d  =  D  ->  (CauFil `  d )  =  (CauFil `  D ) )
13 fveq2 5691 . . . . . . . 8  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  ( MetOpen `  D )
)
14 iscmet.1 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
1513, 14syl6eqr 2493 . . . . . . 7  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  J )
1615oveq1d 6106 . . . . . 6  |-  ( d  =  D  ->  (
( MetOpen `  d )  fLim  f )  =  ( J  fLim  f )
)
1716neeq1d 2621 . . . . 5  |-  ( d  =  D  ->  (
( ( MetOpen `  d
)  fLim  f )  =/=  (/)  <->  ( J  fLim  f )  =/=  (/) ) )
1812, 17raleqbidv 2931 . . . 4  |-  ( d  =  D  ->  ( A. f  e.  (CauFil `  d ) ( (
MetOpen `  d )  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
1918elrab 3117 . . 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719   _Vcvv 2972   (/)c0 3637   ` cfv 5418  (class class class)co 6091   Metcme 17802   MetOpencmopn 17806    fLim cflim 19507  CauFilccfil 20763   CMetcms 20765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-cmet 20768
This theorem is referenced by:  cmetcvg  20796  cmetmet  20797  iscmet3  20804  cmetss  20825  equivcmet  20826  relcmpcmet  20827  cmetcusp1OLD  20863  cmetcusp1  20864
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