Theorem flimtop 23690

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 2731	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	elflim2 23689	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽) ∧ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simplbi 497	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽))
4	3	simp1d 1141	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2105   ⊆ wss 3948  𝒫 cpw 4602  {csn 4628  ∪ cuni 4908  ran crn 5677  ‘cfv 6543  (class class class)co 7412  Topctop 22616  neicnei 22822  Filcfil 23570   fLim cflim 23659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-top 22617  df-flim 23664

This theorem is referenced by:  flimfil  23694  flimtopon  23695  flimss1  23698  flimclsi  23703  hausflimlem  23704  flimsncls  23711  cnpflfi  23724  flimfcls  23751  flimfnfcls  23753