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Theorem cfilss 23068
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 794 . 2 |- ( ( ( D e. ( *Met ` X ) /\ F e. (CauFil ` D ) ) /\ ( G e. ( Fil ` X ) /\ F C_ G ) ) -> G e. ( Fil ` X ) )
2 simprr 796 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ F e. (CauFil ` D ) ) /\ ( G e. ( Fil ` X ) /\ F C_ G ) ) -> F C_ G )
3 iscfil 23063 . . . . 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 654 . . . 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 481 . . 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 3667 . . . 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 2963 . . 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 65 . 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 23063 . . 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 762 . 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 957 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 196   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   X. cxp 5112   "cima 5117   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR+crp 11832   [,)cico 12177   *Metcxmt 19731   Filcfil 21649  CauFilccfil 23050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-xmet 19739  df-cfil 23053
This theorem is referenced by: (None)
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