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Theorem flimelbas 17679
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 17675 . . 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 450 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( A  e.  X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
43simpld 445 1  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   {csn 3653   U.cuni 3843   ran crn 4706   ` cfv 5271  (class class class)co 5874   Topctop 16647   neicnei 16850   Filcfil 17556    fLim cflim 17645
This theorem is referenced by:  flimfil  17680  flimss2  17683  flimss1  17684  flimclsi  17689  hausflimi  17691  flimsncls  17697  cnpflfi  17710  cnflf  17713  cnflf2  17714  flimcfil  18755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-flim 17650
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