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Theorem cfilss 19094
Description: A filter finer than a Cauchy filter is Cauchy. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
cfilss  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  G  e.  (CauFil `  D )
)

Proof of Theorem cfilss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 733 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  G  e.  ( Fil `  X
) )
2 simprr 734 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  F  C_  G )
3 iscfil 19089 . . . . 5  |-  ( 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 ) ) ) )
43simplbda 608 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  (CauFil `  D ) )  ->  A. x  e.  RR+  E. y  e.  F  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) )
54adantr 452 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  A. x  e.  RR+  E. y  e.  F  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) )
6 ssrexv 3351 . . . 4  |-  ( F 
C_  G  ->  ( E. y  e.  F  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x )  ->  E. y  e.  G  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )
76ralimdv 2728 . . 3  |-  ( F 
C_  G  ->  ( A. x  e.  RR+  E. y  e.  F  ( D " ( y  X.  y
) )  C_  (
0 [,) x )  ->  A. x  e.  RR+  E. y  e.  G  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x ) ) )
82, 5, 7sylc 58 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  A. x  e.  RR+  E. y  e.  G  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) )
9 iscfil 19089 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( G  e.  (CauFil `  D
)  <->  ( G  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  G  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
109ad2antrr 707 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  ( G  e.  (CauFil `  D
)  <->  ( G  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  G  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
111, 8, 10mpbir2and 889 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  G  e.  (CauFil `  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   A.wral 2649   E.wrex 2650    C_ wss 3263    X. cxp 4816   "cima 4821   ` cfv 5394  (class class class)co 6020   0cc0 8923   RR+crp 10544   [,)cico 10850   * Metcxmt 16612   Filcfil 17798  CauFilccfil 19076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-map 6956  df-xr 9057  df-xmet 16619  df-cfil 19079
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