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Theorem iscmet 18726
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 5571 . 2  |-  ( D  e.  ( CMet `  X
)  ->  X  e.  _V )
2 elfvex 5571 . . 3  |-  ( D  e.  ( Met `  X
)  ->  X  e.  _V )
32adantr 451 . 2  |-  ( ( D  e.  ( Met `  X )  /\  A. f  e.  (CauFil `  D
) ( J  fLim  f )  =/=  (/) )  ->  X  e.  _V )
4 fveq2 5541 . . . . . 6  |-  ( x  =  X  ->  ( Met `  x )  =  ( Met `  X
) )
5 rabeq 2795 . . . . . 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 15 . . . . 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 18699 . . . . 5  |-  CMet  =  ( x  e.  _V  |->  { d  e.  ( Met `  x )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
8 fvex 5555 . . . . . 6  |-  ( Met `  X )  e.  _V
98rabex 4181 . . . . 5  |-  { d  e.  ( Met `  X
)  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) }  e.  _V
106, 7, 9fvmpt 5618 . . . 4  |-  ( X  e.  _V  ->  ( CMet `  X )  =  { d  e.  ( Met `  X )  |  A. f  e.  (CauFil `  d )
( ( MetOpen `  d
)  fLim  f )  =/=  (/) } )
1110eleq2d 2363 . . 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 5541 . . . . 5  |-  ( d  =  D  ->  (CauFil `  d )  =  (CauFil `  D ) )
13 fveq2 5541 . . . . . . . 8  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  ( MetOpen `  D )
)
14 iscmet.1 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
1513, 14syl6eqr 2346 . . . . . . 7  |-  ( d  =  D  ->  ( MetOpen
`  d )  =  J )
1615oveq1d 5889 . . . . . 6  |-  ( d  =  D  ->  (
( MetOpen `  d )  fLim  f )  =  ( J  fLim  f )
)
1716neeq1d 2472 . . . . 5  |-  ( d  =  D  ->  (
( ( MetOpen `  d
)  fLim  f )  =/=  (/)  <->  ( J  fLim  f )  =/=  (/) ) )
1812, 17raleqbidv 2761 . . . 4  |-  ( d  =  D  ->  ( A. f  e.  (CauFil `  d ) ( (
MetOpen `  d )  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) )
1918elrab 2936 . . 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 252 . 2  |-  ( X  e.  _V  ->  ( D  e.  ( CMet `  X )  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( J  fLim  f
)  =/=  (/) ) ) )
211, 3, 20pm5.21nii 342 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801   (/)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:  cmetcvg  18727  cmetmet  18728  iscmet3  18735  cmetss  18756  equivcmet  18757  relcmpcmet  18758
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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