Theorem elflim2 23352

Description: The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
flimval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
elflim2	⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Proof of Theorem elflim2

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

Step	Hyp	Ref	Expression
1		anass 469	. 2 ⊢ ((((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
2		df-3an 1089	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋))
3	2	anbi1i 624	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) ↔ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
4		df-flim 23327	. . . 4 ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
5	4	elmpocl 7600	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
6		flimval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
7	6	flimval 23351	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})
8	7	eleq2d 2818	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)}))
9		sneq 4601	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	fveq2d 6851	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝐴}))
11	10	sseq1d 3978	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ↔ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
12	11	anbi1d 630	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)))
13	12	biancomd 464	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
14	13	elrab 3648	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
15		an12 643	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
16	14, 15	bitri 274	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
17	8, 16	bitrdi 286	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
18	5, 17	biadanii 820	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
19	1, 3, 18	3bitr4ri 303	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3405   ⊆ wss 3913  𝒫 cpw 4565  {csn 4591  ∪ cuni 4870  ran crn 5639  ‘cfv 6501  (class class class)co 7362  Topctop 22279  neicnei 22485  Filcfil 23233   fLim cflim 23322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-top 22280  df-flim 23327

This theorem is referenced by:  flimtop  23353  flimneiss  23354  flimelbas  23356  flimfil  23357  elflim  23359