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Theorem flimfil 20970
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 20965 . . . . 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 461 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil  /\  F  C_ 
~P X ) )
43simp2d 1018 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  U.
ran  Fil )
5 filunirn 20883 . . 3  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
64, 5sylib 199 . 2  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. F
) )
73simp3d 1019 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  F  C_  ~P X )
8 sspwuni 4385 . . . . 5  |-  ( F 
C_  ~P X  <->  U. F  C_  X )
97, 8sylib 199 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  U. F  C_  X )
10 flimneiss 20967 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A } )  C_  F )
11 flimtop 20966 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
121topopn 19922 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
1311, 12syl 17 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  J )
141flimelbas 20969 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
15 opnneip 20121 . . . . . . 7  |-  ( ( J  e.  Top  /\  X  e.  J  /\  A  e.  X )  ->  X  e.  ( ( nei `  J ) `
 { A }
) )
1611, 13, 14, 15syl3anc 1264 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  ( ( nei `  J
) `  { A } ) )
1710, 16sseldd 3465 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  X  e.  F )
18 elssuni 4245 . . . . 5  |-  ( X  e.  F  ->  X  C_ 
U. F )
1917, 18syl 17 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  X  C_  U. F
)
209, 19eqssd 3481 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  U. F  =  X )
2120fveq2d 5881 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( Fil ` 
U. F )  =  ( Fil `  X
) )
226, 21eleqtrd 2512 1  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    C_ wss 3436   ~Pcpw 3979   {csn 3996   U.cuni 4216   ran crn 4850   ` cfv 5597  (class class class)co 6301   Topctop 19903   neicnei 20099   Filcfil 20846    fLim cflim 20935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-fbas 18954  df-top 19907  df-nei 20100  df-fil 20847  df-flim 20940
This theorem is referenced by:  flimtopon  20971  flimss1  20974  flimclsi  20979  hausflimlem  20980  flimsncls  20987  cnpflfi  21000  flimfcls  21027  flimcfil  22269
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