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Theorem cfilss 20914
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 755 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  G  e.  ( Fil `  X
) )
2 simprr 756 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  F  C_  G )
3 iscfil 20909 . . . . 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 624 . . . 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 465 . . 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 3526 . . . 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 2834 . . 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 60 . 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 20909 . . 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 725 . 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 913 1  |-  ( ( ( D  e.  ( *Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  G  e.  (CauFil `  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758   A.wral 2799   E.wrex 2800    C_ wss 3437    X. cxp 4947   "cima 4952   ` cfv 5527  (class class class)co 6201   0cc0 9394   RR+crp 11103   [,)cico 11414   *Metcxmt 17927   Filcfil 19551  CauFilccfil 20896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-xr 9534  df-xmet 17936  df-cfil 20899
This theorem is referenced by: (None)
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