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Theorem cfiluexsm 20661
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 20659 . . . 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 1016 . 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 3531 . . . . 5  |-  ( v  =  V  ->  (
( a  X.  a
)  C_  v  <->  ( a  X.  a )  C_  V
) )
54rexbidv 2978 . . . 4  |-  ( v  =  V  ->  ( E. a  e.  F  ( a  X.  a
)  C_  v  <->  E. a  e.  F  ( a  X.  a )  C_  V
) )
65rspcv 3215 . . 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 1019 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    C_ wss 3481    X. cxp 5003   ` cfv 5594   fBascfbas 18276  UnifOncust 20570  CauFiluccfilu 20657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ust 20571  df-cfilu 20658
This theorem is referenced by:  fmucnd  20663  cfilucfilOLD  20940  cfilucfil  20941
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