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Theorem cfilss 18696
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 732 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  G  e.  ( Fil `  X
) )
2 simprr 733 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  F  C_  G )
3 iscfil 18691 . . . . 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 607 . . . 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 451 . . 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 3238 . . . 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 2622 . . 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 56 . 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 18691 . . 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 706 . 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 888 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 176    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152    X. cxp 4687   "cima 4692   ` cfv 5255  (class class class)co 5858   0cc0 8737   RR+crp 10354   [,)cico 10658   * Metcxmt 16369   Filcfil 17540  CauFilccfil 18678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-xr 8871  df-xmet 16373  df-cfil 18681
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