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Theorem cfiluexsm 19870
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 19868 . . . 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 624 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `  U ) )  ->  A. v  e.  U  E. a  e.  F  ( a  X.  a
)  C_  v )
323adant3 1008 . 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 3383 . . . . 5  |-  ( v  =  V  ->  (
( a  X.  a
)  C_  v  <->  ( a  X.  a )  C_  V
) )
54rexbidv 2741 . . . 4  |-  ( v  =  V  ->  ( E. a  e.  F  ( a  X.  a
)  C_  v  <->  E. a  e.  F  ( a  X.  a )  C_  V
) )
65rspcv 3074 . . 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 1011 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721    C_ wss 3333    X. cxp 4843   ` cfv 5423   fBascfbas 17809  UnifOncust 19779  CauFiluccfilu 19866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431  df-ust 19780  df-cfilu 19867
This theorem is referenced by:  fmucnd  19872  cfilucfilOLD  20149  cfilucfil  20150
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