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Theorem flimfil 17680
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimuni.1  |-  X  = 
U. J
Assertion
Ref Expression
flimfil  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  X ) )

Proof of Theorem flimfil
StepHypRef Expression
1 flimuni.1 . . . . . 6  |-  X  = 
U. J
21elflim2 17675 . . . . 5  |-  ( 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 )
) )
32simplbi 446 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil  /\  F  C_ 
~P X ) )
43simp2d 968 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  U.
ran  Fil )
5 filunirn 17593 . . 3  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
64, 5sylib 188 . 2  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. F
) )
73simp3d 969 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  F  C_  ~P X )
8 sspwuni 4003 . . . . 5  |-  ( F 
C_  ~P X  <->  U. F  C_  X )
97, 8sylib 188 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  U. F  C_  X )
10 flimneiss 17677 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A } )  C_  F )
11 flimtop 17676 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
121topopn 16668 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
1311, 12syl 15 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  J )
141flimelbas 17679 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
15 opnneip 16872 . . . . . . 7  |-  ( ( J  e.  Top  /\  X  e.  J  /\  A  e.  X )  ->  X  e.  ( ( nei `  J ) `
 { A }
) )
1611, 13, 14, 15syl3anc 1182 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  ( ( nei `  J
) `  { A } ) )
1710, 16sseldd 3194 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  F )
18 elssuni 3871 . . . . 5  |-  ( X  e.  F  ->  X  C_ 
U. F )
1917, 18syl 15 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  X  C_  U. F
)
209, 19eqssd 3209 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  U. F  =  X )
2120fveq2d 5545 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( Fil ` 
U. F )  =  ( Fil `  X
) )
226, 21eleqtrd 2372 1  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  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:  flimtopon  17681  flimss1  17684  flimclsi  17689  hausflimlem  17690  flimsncls  17697  cnpflfi  17710  flimfcls  17737  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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-nei 16851  df-fbas 17536  df-fil 17557  df-flim 17650
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