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Theorem flimfil 20297
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 20292 . . . . 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 460 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil  /\  F  C_ 
~P X ) )
43simp2d 1009 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  U.
ran  Fil )
5 filunirn 20210 . . 3  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
64, 5sylib 196 . 2  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. F
) )
73simp3d 1010 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  F  C_  ~P X )
8 sspwuni 4411 . . . . 5  |-  ( F 
C_  ~P X  <->  U. F  C_  X )
97, 8sylib 196 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  U. F  C_  X )
10 flimneiss 20294 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A } )  C_  F )
11 flimtop 20293 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
121topopn 19222 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
1311, 12syl 16 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  J )
141flimelbas 20296 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
15 opnneip 19426 . . . . . . 7  |-  ( ( J  e.  Top  /\  X  e.  J  /\  A  e.  X )  ->  X  e.  ( ( nei `  J ) `
 { A }
) )
1611, 13, 14, 15syl3anc 1228 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  ( ( nei `  J
) `  { A } ) )
1710, 16sseldd 3505 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  F )
18 elssuni 4275 . . . . 5  |-  ( X  e.  F  ->  X  C_ 
U. F )
1917, 18syl 16 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  X  C_  U. F
)
209, 19eqssd 3521 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  U. F  =  X )
2120fveq2d 5870 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( Fil ` 
U. F )  =  ( Fil `  X
) )
226, 21eleqtrd 2557 1  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   ~Pcpw 4010   {csn 4027   U.cuni 4245   ran crn 5000   ` cfv 5588  (class class class)co 6285   Topctop 19201   neicnei 19404   Filcfil 20173    fLim cflim 20262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-fbas 18227  df-top 19206  df-nei 19405  df-fil 20174  df-flim 20267
This theorem is referenced by:  flimtopon  20298  flimss1  20301  flimclsi  20306  hausflimlem  20307  flimsncls  20314  cnpflfi  20327  flimfcls  20354  flimcfil  21579
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