Theorem flimtop 24105

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

Step	Hyp	Ref	Expression
1		eqid 2770	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	elflim2 24104	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽) ∧ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simplbi 501	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽))
4	3	simp1d 1158	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2150   ⊆ wss 3913  𝒫 cpw 4567  {csn 4594  ∪ cuni 4877  ran crn 5666  ‘cfv 6540  (class class class)co 7414  Topctop 23033  neicnei 23237  Filcfil 23985   fLim cflim 24074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-top 23034  df-flim 24079

This theorem is referenced by:  flimfil  24109  flimtopon  24110  flimss1  24113  flimclsi  24118  hausflimlem  24119  flimsncls  24126  cnpflfi  24139  flimfcls  24166  flimfnfcls  24168