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Theorem cfilfval 18706
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 5567 . . . 4  |-  ( * Met `  X ) 
C_  U. ran  * Met
21sseli 3189 . . 3  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
3 dmeq 4895 . . . . . . 7  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 4897 . . . . . 6  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
54fveq2d 5545 . . . . 5  |-  ( d  =  D  ->  ( Fil `  dom  dom  d
)  =  ( Fil `  dom  dom  D )
)
6 imaeq1 5023 . . . . . . . 8  |-  ( d  =  D  ->  (
d " ( y  X.  y ) )  =  ( D "
( y  X.  y
) ) )
76sseq1d 3218 . . . . . . 7  |-  ( d  =  D  ->  (
( d " (
y  X.  y ) )  C_  ( 0 [,) x )  <->  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )
87rexbidv 2577 . . . . . 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 2576 . . . . 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 2796 . . . 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 18697 . . . 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 5555 . . . . 5  |-  ( Fil `  dom  dom  D )  e.  _V
1312rabex 4181 . . . 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 5618 . . 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 15 . 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 17916 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  X  =  dom  dom  D )
1716fveq2d 5545 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( Fil `  X )  =  ( Fil `  dom  dom 
D ) )
18 rabeq 2795 . . 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 15 . 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 2331 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   U.cuni 3843    X. cxp 4703   dom cdm 4705   ran crn 4706   "cima 4708   ` cfv 5271  (class class class)co 5874   0cc0 8753   RR+crp 10370   [,)cico 10674   * Metcxmt 16385   Filcfil 17556  CauFilccfil 18694
This theorem is referenced by:  iscfil  18707
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  ax-un 4528  ax-cnex 8809  ax-resscn 8810
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-pw 3640  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-xr 8887  df-xmet 16389  df-cfil 18697
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