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Theorem cfiluexsm 21305
Description: For a Cauchy filter base and any entourage  V, there is an element of the filter small in  V. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Assertion
Ref Expression
cfiluexsm  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U )  /\  V  e.  U )  ->  E. a  e.  F  ( a  X.  a
)  C_  V )
Distinct variable groups:    F, a    V, a
Allowed substitution hints:    U( a)    X( a)

Proof of Theorem cfiluexsm
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 iscfilu 21303 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  (CauFilu `  U )  <->  ( F  e.  ( fBas `  X
)  /\  A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v
) ) )
21simplbda 630 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  A. v  e.  U  E. a  e.  F  ( a  X.  a
)  C_  v )
323adant3 1028 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U )  /\  V  e.  U )  ->  A. v  e.  U  E. a  e.  F  ( a  X.  a
)  C_  v )
4 sseq2 3454 . . . . 5  |-  ( v  =  V  ->  (
( a  X.  a
)  C_  v  <->  ( a  X.  a )  C_  V
) )
54rexbidv 2901 . . . 4  |-  ( v  =  V  ->  ( E. a  e.  F  ( a  X.  a
)  C_  v  <->  E. a  e.  F  ( a  X.  a )  C_  V
) )
65rspcv 3146 . . 3  |-  ( V  e.  U  ->  ( A. v  e.  U  E. a  e.  F  ( a  X.  a
)  C_  v  ->  E. a  e.  F  ( a  X.  a ) 
C_  V ) )
763ad2ant3 1031 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U )  /\  V  e.  U )  ->  ( A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v  ->  E. a  e.  F  ( a  X.  a
)  C_  V )
)
83, 7mpd 15 1  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U )  /\  V  e.  U )  ->  E. a  e.  F  ( a  X.  a
)  C_  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738    C_ wss 3404    X. cxp 4832   ` cfv 5582   fBascfbas 18958  UnifOncust 21214  CauFiluccfilu 21301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-ust 21215  df-cfilu 21302
This theorem is referenced by:  fmucnd  21307  cfilucfil  21574
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