Theorem flimfil 23952

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 23947	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simplbi 497	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋))
4	3	simp2d 1149	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ ∪ ran Fil)
5		filunirn 23865	. . 3 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
6	4, 5	sylib 219	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐹))
7	3	simp3d 1150	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
8		sspwuni 5029	. . . . 5 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋)
9	7, 8	sylib 219	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 ⊆ 𝑋)
10		flimneiss 23949	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
11		flimtop 23948	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
12	1	topopn 22889	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
13	11, 12	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐽)
14	1	flimelbas 23951	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋)
15		opnneip 23102	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
16	11, 13, 14, 15	syl3anc 1379	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
17	10, 16	sseldd 3916	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐹)
18		elssuni 4869	. . . . 5 ⊢ (𝑋 ∈ 𝐹 → 𝑋 ⊆ ∪ 𝐹)
19	17, 18	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ⊆ ∪ 𝐹)
20	9, 19	eqssd 3932	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 = 𝑋)
21	20	fveq2d 6831	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘∪ 𝐹) = (Fil‘𝑋))
22	6, 21	eleqtrd 2841	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  𝒫 cpw 4529  {csn 4555  ∪ cuni 4838  ran crn 5619  ‘cfv 6485  (class class class)co 7356  Topctop 22876  neicnei 23080  Filcfil 23828   fLim cflim 23917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-fbas 21344  df-top 22877  df-nei 23081  df-fil 23829  df-flim 23922

This theorem is referenced by:  flimtopon  23953  flimss1  23956  flimclsi  23961  hausflimlem  23962  flimsncls  23969  cnpflfi  23982  flimfcls  24009  flimcfil  25299