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Theorem flimtop 20194
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 2460 . . . 4  |-  U. J  =  U. J
21elflim2 20193 . . 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 460 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil  /\  F  C_ 
~P U. J ) )
43simp1d 1003 1  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    e. wcel 1762    C_ wss 3469   ~Pcpw 4003   {csn 4020   U.cuni 4238   ran crn 4993   ` cfv 5579  (class class class)co 6275   Topctop 19154   neicnei 19357   Filcfil 20074    fLim cflim 20163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-top 19159  df-flim 20168
This theorem is referenced by:  flimfil  20198  flimtopon  20199  flimss1  20202  flimclsi  20207  hausflimlem  20208  flimsncls  20215  cnpflfi  20228  flimfcls  20255  flimfnfcls  20257
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