Theorem flimval 23953

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 22896	. . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3	2	adantr 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → 𝑋 ∈ 𝐽)
4		rabexg 5272	. . 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 483	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑗 = 𝐽)
7	6	unieqd 4858	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = ∪ 𝐽)
8	7, 1	eqtr4di 2793	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∪ 𝑗 = 𝑋)
9	6	fveq2d 6838	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (nei‘𝑗) = (nei‘𝐽))
10	9	fveq1d 6836	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
11		simpr 485	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
12	10, 11	sseq12d 3955	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ↔ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))
13	8	pweqd 4553	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
14	11, 13	sseq12d 3955	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (𝑓 ⊆ 𝒫 ∪ 𝑗 ↔ 𝐹 ⊆ 𝒫 𝑋))
15	12, 14	anbi12d 638	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ((((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗) ↔ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)))
16	8, 15	rabeqbidv 3410	. . 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 23929	. . 3 ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
18	16, 17	ovmpoga 7517	. 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 1470	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432   ⊆ wss 3890  𝒫 cpw 4536  {csn 4562  ∪ cuni 4845  ran crn 5626  ‘cfv 6492  (class class class)co 7363  Topctop 22883  neicnei 23087  Filcfil 23835   fLim cflim 23924

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 2712  ax-sep 5225  ax-pr 5369

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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-top 22884  df-flim 23929

This theorem is referenced by:  elflim2  23954