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Theorem flimfil 18032
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 18027 . . . . 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 448 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil  /\  F  C_ 
~P X ) )
43simp2d 971 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  U.
ran  Fil )
5 filunirn 17945 . . 3  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
64, 5sylib 190 . 2  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. F
) )
73simp3d 972 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  F  C_  ~P X )
8 sspwuni 4201 . . . . 5  |-  ( F 
C_  ~P X  <->  U. F  C_  X )
97, 8sylib 190 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  U. F  C_  X )
10 flimneiss 18029 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A } )  C_  F )
11 flimtop 18028 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
121topopn 17010 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
1311, 12syl 16 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  J )
141flimelbas 18031 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
15 opnneip 17214 . . . . . . 7  |-  ( ( J  e.  Top  /\  X  e.  J  /\  A  e.  X )  ->  X  e.  ( ( nei `  J ) `
 { A }
) )
1611, 13, 14, 15syl3anc 1185 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  ( ( nei `  J
) `  { A } ) )
1710, 16sseldd 3335 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  F )
18 elssuni 4067 . . . . 5  |-  ( X  e.  F  ->  X  C_ 
U. F )
1917, 18syl 16 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  X  C_  U. F
)
209, 19eqssd 3351 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  U. F  =  X )
2120fveq2d 5761 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( Fil ` 
U. F )  =  ( Fil `  X
) )
226, 21eleqtrd 2518 1  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    C_ wss 3306   ~Pcpw 3823   {csn 3838   U.cuni 4039   ran crn 4908   ` cfv 5483  (class class class)co 6110   Topctop 16989   neicnei 17192   Filcfil 17908    fLim cflim 17997
This theorem is referenced by:  flimtopon  18033  flimss1  18036  flimclsi  18041  hausflimlem  18042  flimsncls  18049  cnpflfi  18062  flimfcls  18089  flimcfil  19297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-fbas 16730  df-top 16994  df-nei 17193  df-fil 17909  df-flim 18002
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