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Theorem cfilss 18712
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 18707 . . . . 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 3251 . . . 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 2635 . . 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 18707 . . 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 1696   A.wral 2556   E.wrex 2557    C_ wss 3165    X. cxp 4703   "cima 4708   ` cfv 5271  (class class class)co 5874   0cc0 8753   RR+crp 10370   [,)cico 10674   * Metcxmt 16385   Filcfil 17556  CauFilccfil 18694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-xr 8887  df-xmet 16389  df-cfil 18697
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