Theorem flimfil 24288

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 24283	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simplbi 502	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋))
4	3	simp2d 1161	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ ∪ ran Fil)
5		filunirn 24201	. . 3 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
6	4, 5	sylib 221	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐹))
7	3	simp3d 1162	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
8		sspwuni 5059	. . . . 5 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋)
9	7, 8	sylib 221	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 ⊆ 𝑋)
10		flimneiss 24285	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
11		flimtop 24284	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
12	1	topopn 23224	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
13	11, 12	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐽)
14	1	flimelbas 24287	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋)
15		opnneip 23437	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
16	11, 13, 14, 15	syl3anc 1398	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ ((nei‘𝐽)‘{𝐴}))
17	10, 16	sseldd 3909	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ∈ 𝐹)
18		elssuni 4898	. . . . 5 ⊢ (𝑋 ∈ 𝐹 → 𝑋 ⊆ ∪ 𝐹)
19	17, 18	syl 18	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝑋 ⊆ ∪ 𝐹)
20	9, 19	eqssd 3925	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 = 𝑋)
21	20	fveq2d 6889	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (Fil‘∪ 𝐹) = (Fil‘𝑋))
22	6, 21	eleqtrd 2862	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895  𝒫 cpw 4556  {csn 4583  ∪ cuni 4866  ran crn 5651  ‘cfv 6538  (class class class)co 7420  Topctop 23211  neicnei 23415  Filcfil 24164   fLim cflim 24253

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-fbas 21675  df-top 23212  df-nei 23416  df-fil 24165  df-flim 24258

This theorem is used by:  flimtopon  24289  flimss1  24292  flimclsi  24297  hausflimlem  24298  flimsncls  24305  cnpflfi  24318  flimfcls  24345  flimcfil  25635