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Theorem cfilfval 22289
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 5915 . . . 4  |-  ( *Met `  X ) 
C_  U. ran  *Met
21sseli 3440 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
3 dmeq 5057 . . . . . . 7  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5059 . . . . . 6  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
54fveq2d 5896 . . . . 5  |-  ( d  =  D  ->  ( Fil `  dom  dom  d
)  =  ( Fil `  dom  dom  D )
)
6 imaeq1 5185 . . . . . . . 8  |-  ( d  =  D  ->  (
d " ( y  X.  y ) )  =  ( D "
( y  X.  y
) ) )
76sseq1d 3471 . . . . . . 7  |-  ( d  =  D  ->  (
( d " (
y  X.  y ) )  C_  ( 0 [,) x )  <->  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )
87rexbidv 2913 . . . . . 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 2839 . . . . 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 3052 . . . 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 22280 . . . 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 5902 . . . . 5  |-  ( Fil `  dom  dom  D )  e.  _V
1312rabex 4571 . . . 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 5976 . . 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 17 . 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 21405 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  =  dom  dom  D )
1716fveq2d 5896 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( Fil `  X )  =  ( Fil `  dom  dom 
D ) )
18 rabeq 3050 . . 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 17 . 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 2499 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 setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   {crab 2753    C_ wss 3416   U.cuni 4212    X. cxp 4854   dom cdm 4856   ran crn 4857   "cima 4859   ` cfv 5605  (class class class)co 6320   0cc0 9570   RR+crp 11336   [,)cico 11671   *Metcxmt 19010   Filcfil 20915  CauFilccfil 22277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-map 7505  df-xr 9710  df-xmet 19018  df-cfil 22280
This theorem is referenced by:  iscfil  22290
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