MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfilfval Structured version   Unicode version

Theorem cfilfval 20877
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 5798 . . . 4  |-  ( *Met `  X ) 
C_  U. ran  *Met
21sseli 3436 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
3 dmeq 5124 . . . . . . 7  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5126 . . . . . 6  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
54fveq2d 5779 . . . . 5  |-  ( d  =  D  ->  ( Fil `  dom  dom  d
)  =  ( Fil `  dom  dom  D )
)
6 imaeq1 5248 . . . . . . . 8  |-  ( d  =  D  ->  (
d " ( y  X.  y ) )  =  ( D "
( y  X.  y
) ) )
76sseq1d 3467 . . . . . . 7  |-  ( d  =  D  ->  (
( d " (
y  X.  y ) )  C_  ( 0 [,) x )  <->  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )
87rexbidv 2816 . . . . . 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 2815 . . . . 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 3049 . . . 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 20868 . . . 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 5785 . . . . 5  |-  ( Fil `  dom  dom  D )  e.  _V
1312rabex 4527 . . . 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 5859 . . 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 20012 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  =  dom  dom  D )
1716fveq2d 5779 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( Fil `  X )  =  ( Fil `  dom  dom 
D ) )
18 rabeq 3048 . . 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 2493 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 1370    e. wcel 1757   A.wral 2792   E.wrex 2793   {crab 2796    C_ wss 3412   U.cuni 4175    X. cxp 4922   dom cdm 4924   ran crn 4925   "cima 4927   ` cfv 5502  (class class class)co 6176   0cc0 9369   RR+crp 11078   [,)cico 11389   *Metcxmt 17896   Filcfil 19520  CauFilccfil 20865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-map 7302  df-xr 9509  df-xmet 17905  df-cfil 20868
This theorem is referenced by:  iscfil  20878
  Copyright terms: Public domain W3C validator