Theorem flimval 23987

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 22928	. . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3	2	adantr 480	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → 𝑋 ∈ 𝐽)
4		rabexg 5343	. . 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 482	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑗 = 𝐽)
7	6	unieqd 4925	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = ∪ 𝐽)
8	7, 1	eqtr4di 2793	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = 𝑋)
9	6	fveq2d 6911	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (nei‘𝑗) = (nei‘𝐽))
10	9	fveq1d 6909	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
11		simpr 484	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
12	10, 11	sseq12d 4029	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ↔ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))
13	8	pweqd 4622	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
14	11, 13	sseq12d 4029	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (𝑓 ⊆ 𝒫 ∪ 𝑗 ↔ 𝐹 ⊆ 𝒫 𝑋))
15	12, 14	anbi12d 632	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗) ↔ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)))
16	8, 15	rabeqbidv 3452	. . 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 23963	. . 3 ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
18	16, 17	ovmpoga 7587	. 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 1461	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912  ran crn 5690  ‘cfv 6563  (class class class)co 7431  Topctop 22915  neicnei 23121  Filcfil 23869   fLim cflim 23958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-top 22916  df-flim 23963

This theorem is referenced by:  elflim2  23988