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Theorem cfilss 21875
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 754 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  G  e.  ( Fil `  X
) )
2 simprr 755 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  F  e.  (CauFil `  D ) )  /\  ( G  e.  ( Fil `  X
)  /\  F  C_  G
) )  ->  F  C_  G )
3 iscfil 21870 . . . . 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 622 . . . 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 463 . . 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 3551 . . . 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 2864 . . 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 21870 . . 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 723 . 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 920 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 367    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461    X. cxp 4986   "cima 4991   ` cfv 5570  (class class class)co 6270   0cc0 9481   RR+crp 11221   [,)cico 11534   *Metcxmt 18598   Filcfil 20512  CauFilccfil 21857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-xr 9621  df-xmet 18607  df-cfil 21860
This theorem is referenced by: (None)
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