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Theorem flimtop 21709
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 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)

Proof of Theorem flimtop
StepHypRef Expression
1 eqid 2621 . . . 4 𝐽 = 𝐽
21elflim2 21708 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝐽) ∧ (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
32simplbi 476 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝐽))
43simp1d 1071 1 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1987  wss 3560  𝒫 cpw 4136  {csn 4155   cuni 4409  ran crn 5085  cfv 5857  (class class class)co 6615  Topctop 20638  neicnei 20841  Filcfil 21589   fLim cflim 21678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-top 20639  df-flim 21683
This theorem is referenced by:  flimfil  21713  flimtopon  21714  flimss1  21717  flimclsi  21722  hausflimlem  21723  flimsncls  21730  cnpflfi  21743  flimfcls  21770  flimfnfcls  21772
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