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Theorem flimfil 17664
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 17659 . . . . 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 17577 . . 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 3987 . . . . 5  |-  ( F 
C_  ~P X  <->  U. F  C_  X )
97, 8sylib 188 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  U. F  C_  X )
10 flimneiss 17661 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A } )  C_  F )
11 flimtop 17660 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
121topopn 16652 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
1311, 12syl 15 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  J )
141flimelbas 17663 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
15 opnneip 16856 . . . . . . 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 3181 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  F )
18 elssuni 3855 . . . . 5  |-  ( X  e.  F  ->  X  C_ 
U. F )
1917, 18syl 15 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  X  C_  U. F
)
209, 19eqssd 3196 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  U. F  =  X )
2120fveq2d 5529 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( Fil ` 
U. F )  =  ( Fil `  X
) )
226, 21eleqtrd 2359 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 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   ran crn 4690   ` cfv 5255  (class class class)co 5858   Topctop 16631   neicnei 16834   Filcfil 17540    fLim cflim 17629
This theorem is referenced by:  flimtopon  17665  flimss1  17668  flimclsi  17673  hausflimlem  17674  flimsncls  17681  cnpflfi  17694  flimfcls  17721  flimcfil  18739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-nei 16835  df-fbas 17520  df-fil 17541  df-flim 17634
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