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Theorem cfilfval 21869
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 5871 . . . 4  |-  ( *Met `  X ) 
C_  U. ran  *Met
21sseli 3485 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
3 dmeq 5192 . . . . . . 7  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5194 . . . . . 6  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
54fveq2d 5852 . . . . 5  |-  ( d  =  D  ->  ( Fil `  dom  dom  d
)  =  ( Fil `  dom  dom  D )
)
6 imaeq1 5320 . . . . . . . 8  |-  ( d  =  D  ->  (
d " ( y  X.  y ) )  =  ( D "
( y  X.  y
) ) )
76sseq1d 3516 . . . . . . 7  |-  ( d  =  D  ->  (
( d " (
y  X.  y ) )  C_  ( 0 [,) x )  <->  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )
87rexbidv 2965 . . . . . 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 2893 . . . . 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 3101 . . . 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 21860 . . . 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 5858 . . . . 5  |-  ( Fil `  dom  dom  D )  e.  _V
1312rabex 4588 . . . 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 5931 . . 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 21004 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  =  dom  dom  D )
1716fveq2d 5852 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( Fil `  X )  =  ( Fil `  dom  dom 
D ) )
18 rabeq 3100 . . 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 2498 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 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808    C_ wss 3461   U.cuni 4235    X. cxp 4986   dom cdm 4988   ran crn 4989   "cima 4991   ` cfv 5570  (class class class)co 6270   0cc0 9481   RR+crp 11221   [,)cico 11534   *Metcxmt 18598   Filcfil 20512  CauFilccfil 21857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-xr 9621  df-xmet 18607  df-cfil 21860
This theorem is referenced by:  iscfil  21870
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