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Theorem cfilfval 19217
Description: The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
cfilfval  |-  ( D  e.  ( * Met `  X )  ->  (CauFil `  D )  =  {
f  e.  ( Fil `  X )  |  A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) } )
Distinct variable groups:    x, y,
f, X    D, f, x, y

Proof of Theorem cfilfval
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 fvssunirn 5754 . . . 4  |-  ( * Met `  X ) 
C_  U. ran  * Met
21sseli 3344 . . 3  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
3 dmeq 5070 . . . . . . 7  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5072 . . . . . 6  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
54fveq2d 5732 . . . . 5  |-  ( d  =  D  ->  ( Fil `  dom  dom  d
)  =  ( Fil `  dom  dom  D )
)
6 imaeq1 5198 . . . . . . . 8  |-  ( d  =  D  ->  (
d " ( y  X.  y ) )  =  ( D "
( y  X.  y
) ) )
76sseq1d 3375 . . . . . . 7  |-  ( d  =  D  ->  (
( d " (
y  X.  y ) )  C_  ( 0 [,) x )  <->  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )
87rexbidv 2726 . . . . . 6  |-  ( d  =  D  ->  ( E. y  e.  f 
( d " (
y  X.  y ) )  C_  ( 0 [,) x )  <->  E. y  e.  f  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )
98ralbidv 2725 . . . . 5  |-  ( d  =  D  ->  ( A. x  e.  RR+  E. y  e.  f  ( d " ( y  X.  y ) )  C_  ( 0 [,) x
)  <->  A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x ) ) )
105, 9rabeqbidv 2951 . . . 4  |-  ( d  =  D  ->  { f  e.  ( Fil `  dom  dom  d )  |  A. x  e.  RR+  E. y  e.  f  ( d " ( y  X.  y ) )  C_  ( 0 [,) x
) }  =  {
f  e.  ( Fil `  dom  dom  D )  |  A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x ) } )
11 df-cfil 19208 . . . 4  |- CauFil  =  ( d  e.  U. ran  * Met  |->  { f  e.  ( Fil `  dom  dom  d )  |  A. x  e.  RR+  E. y  e.  f  ( d " ( y  X.  y ) )  C_  ( 0 [,) x
) } )
12 fvex 5742 . . . . 5  |-  ( Fil `  dom  dom  D )  e.  _V
1312rabex 4354 . . . 4  |-  { f  e.  ( Fil `  dom  dom 
D )  |  A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) }  e.  _V
1410, 11, 13fvmpt 5806 . . 3  |-  ( D  e.  U. ran  * Met  ->  (CauFil `  D )  =  { f  e.  ( Fil `  dom  dom  D )  |  A. x  e.  RR+  E. y  e.  f  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) } )
152, 14syl 16 . 2  |-  ( D  e.  ( * Met `  X )  ->  (CauFil `  D )  =  {
f  e.  ( Fil `  dom  dom  D )  |  A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x ) } )
16 xmetdmdm 18365 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  X  =  dom  dom  D )
1716fveq2d 5732 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( Fil `  X )  =  ( Fil `  dom  dom 
D ) )
18 rabeq 2950 . . 3  |-  ( ( Fil `  X )  =  ( Fil `  dom  dom 
D )  ->  { f  e.  ( Fil `  X
)  |  A. x  e.  RR+  E. y  e.  f  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) }  =  { f  e.  ( Fil `  dom  dom 
D )  |  A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) } )
1917, 18syl 16 . 2  |-  ( D  e.  ( * Met `  X )  ->  { f  e.  ( Fil `  X
)  |  A. x  e.  RR+  E. y  e.  f  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) }  =  { f  e.  ( Fil `  dom  dom 
D )  |  A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) } )
2015, 19eqtr4d 2471 1  |-  ( D  e.  ( * Met `  X )  ->  (CauFil `  D )  =  {
f  e.  ( Fil `  X )  |  A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709    C_ wss 3320   U.cuni 4015    X. cxp 4876   dom cdm 4878   ran crn 4879   "cima 4881   ` cfv 5454  (class class class)co 6081   0cc0 8990   RR+crp 10612   [,)cico 10918   * Metcxmt 16686   Filcfil 17877  CauFilccfil 19205
This theorem is referenced by:  iscfil  19218
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-xr 9124  df-xmet 16695  df-cfil 19208
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