Theorem flimfil 22820

Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
flimuni.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
flimfil	⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Proof of Theorem flimfil

Step	Hyp	Ref	Expression
1		flimuni.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2	1	elflim2 22815	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simplbi 501	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋))
4	3	simp2d 1145	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ ∪ ran Fil)
5		filunirn 22733	. . 3 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
6	4, 5	sylib 221	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐹))
7	3	simp3d 1146	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
8		sspwuni 4994	. . . . 5 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋)
9	7, 8	sylib 221	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 ⊆ 𝑋)
10		flimneiss 22817	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
11		flimtop 22816	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
12	1	topopn 21757	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
13	11, 12	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐽)
14	1	flimelbas 22819	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋)
15		opnneip 21970	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
16	11, 13, 14, 15	syl3anc 1373	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
17	10, 16	sseldd 3888	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐹)
18		elssuni 4837	. . . . 5 ⊢ (𝑋 ∈ 𝐹 → 𝑋 ⊆ ∪ 𝐹)
19	17, 18	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ⊆ ∪ 𝐹)
20	9, 19	eqssd 3904	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 = 𝑋)
21	20	fveq2d 6699	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘∪ 𝐹) = (Fil‘𝑋))
22	6, 21	eleqtrd 2833	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ⊆ wss 3853  𝒫 cpw 4499  {csn 4527  ∪ cuni 4805  ran crn 5537  ‘cfv 6358  (class class class)co 7191  Topctop 21744  neicnei 21948  Filcfil 22696   fLim cflim 22785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-fbas 20314  df-top 21745  df-nei 21949  df-fil 22697  df-flim 22790

This theorem is referenced by:  flimtopon  22821  flimss1  22824  flimclsi  22829  hausflimlem  22830  flimsncls  22837  cnpflfi  22850  flimfcls  22877  flimcfil  24165