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Theorem iscfil 21572
Description: The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
iscfil  |-  ( D  e.  ( *Met `  X )  ->  ( F  e.  (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, X, y    x, D, y

Proof of Theorem iscfil
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 cfilfval 21571 . . 3  |-  ( 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 ) } )
21eleq2d 2537 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( F  e.  (CauFil `  D
)  <->  F  e.  { f  e.  ( Fil `  X
)  |  A. x  e.  RR+  E. y  e.  f  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) } ) )
3 rexeq 3064 . . . 4  |-  ( f  =  F  ->  ( E. y  e.  f 
( D " (
y  X.  y ) )  C_  ( 0 [,) x )  <->  E. y  e.  F  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )
43ralbidv 2906 . . 3  |-  ( f  =  F  ->  ( 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 ) ) )
54elrab 3266 . 2  |-  ( F  e.  { f  e.  ( Fil `  X
)  |  A. x  e.  RR+  E. y  e.  f  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) }  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  F  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) )
62, 5syl6bb 261 1  |-  ( D  e.  ( *Met `  X )  ->  ( F  e.  (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    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821    C_ wss 3481    X. cxp 5003   "cima 5008   ` cfv 5594  (class class class)co 6295   0cc0 9504   RR+crp 11232   [,)cico 11543   *Metcxmt 18273   Filcfil 20214  CauFilccfil 21559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-xr 9644  df-xmet 18282  df-cfil 21562
This theorem is referenced by:  iscfil2  21573  cfilfil  21574  cfilss  21577  cfilucfil3OLD  21625  cfilucfil3  21626  cmetcuspOLD  21661  cmetcusp  21662
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