Theorem flimval 21577

Description: The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
flimval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
flimval	⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝑋

Proof of Theorem flimval

Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		flimval.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	topopn 20536	. . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3	2	adantr 480	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → 𝑋 ∈ 𝐽)
4		rabexg 4739	. . 3 ⊢ (𝑋 ∈ 𝐽 → {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ∈ V)
5	3, 4	syl 17	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ∈ V)
6		simpl 472	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑗 = 𝐽)
7	6	unieqd 4382	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = ∪ 𝐽)
8	7, 1	syl6eqr 2662	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = 𝑋)
9	6	fveq2d 6107	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (nei‘𝑗) = (nei‘𝐽))
10	9	fveq1d 6105	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
11		simpr 476	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
12	10, 11	sseq12d 3597	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ↔ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))
13	8	pweqd 4113	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
14	11, 13	sseq12d 3597	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (𝑓 ⊆ 𝒫 ∪ 𝑗 ↔ 𝐹 ⊆ 𝒫 𝑋))
15	12, 14	anbi12d 743	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗) ↔ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)))
16	8, 15	rabeqbidv 3168	. . 3 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)} = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})
17		df-flim 21553	. . 3 ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
18	16, 17	ovmpt2ga 6688	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ∈ V) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})
19	5, 18	mpd3an3 1417	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Topctop 20517  neicnei 20711  Filcfil 21459   fLim cflim 21548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-top 20521  df-flim 21553

This theorem is referenced by:  elflim2  21578