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Theorem flimtop 20972
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
flimtop  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )

Proof of Theorem flimtop
StepHypRef Expression
1 eqid 2423 . . . 4  |-  U. J  =  U. J
21elflim2 20971 . . 3  |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P U. J
)  /\  ( A  e.  U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
32simplbi 462 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil  /\  F  C_ 
~P U. J ) )
43simp1d 1018 1  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    e. wcel 1869    C_ wss 3437   ~Pcpw 3980   {csn 3997   U.cuni 4217   ran crn 4852   ` cfv 5599  (class class class)co 6303   Topctop 19909   neicnei 20105   Filcfil 20852    fLim cflim 20941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-top 19913  df-flim 20946
This theorem is referenced by:  flimfil  20976  flimtopon  20977  flimss1  20980  flimclsi  20985  hausflimlem  20986  flimsncls  20993  cnpflfi  21006  flimfcls  21033  flimfnfcls  21035
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