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Theorem cfiluexsm 21083
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 21081 . . . 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 622 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  A. v  e.  U  E. a  e.  F  ( a  X.  a
)  C_  v )
323adant3 1017 . 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 3463 . . . . 5  |-  ( v  =  V  ->  (
( a  X.  a
)  C_  v  <->  ( a  X.  a )  C_  V
) )
54rexbidv 2917 . . . 4  |-  ( v  =  V  ->  ( E. a  e.  F  ( a  X.  a
)  C_  v  <->  E. a  e.  F  ( a  X.  a )  C_  V
) )
65rspcv 3155 . . 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 1020 . 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 974    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754    C_ wss 3413    X. cxp 4820   ` cfv 5568   fBascfbas 18724  UnifOncust 20992  CauFiluccfilu 21079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-iota 5532  df-fun 5570  df-fn 5571  df-fv 5576  df-ust 20993  df-cfilu 21080
This theorem is referenced by:  fmucnd  21085  cfilucfilOLD  21362  cfilucfil  21363
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