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Theorem flimelbas 17921
Description: A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
flimuni.1  |-  X  = 
U. J
Assertion
Ref Expression
flimelbas  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )

Proof of Theorem flimelbas
StepHypRef Expression
1 flimuni.1 . . . 4  |-  X  = 
U. J
21elflim2 17917 . . 3  |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P X )  /\  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  F )
) )
32simprbi 451 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( A  e.  X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
43simpld 446 1  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3263   ~Pcpw 3742   {csn 3757   U.cuni 3957   ran crn 4819   ` cfv 5394  (class class class)co 6020   Topctop 16881   neicnei 17084   Filcfil 17798    fLim cflim 17887
This theorem is referenced by:  flimfil  17922  flimss2  17925  flimss1  17926  flimclsi  17931  hausflimi  17933  flimsncls  17939  cnpflfi  17952  cnflf  17955  cnflf2  17956  flimcfil  19137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-top 16886  df-flim 17892
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