Theorem fclsopni 24023

Description: An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclsopni	⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅)

Proof of Theorem fclsopni

Dummy variables 𝑜 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclsfil 24018	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
3		fclstopon 24020	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ 𝐹 ∈ (Fil‘∪ 𝐽)))
4	2, 3	mpbird 257	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		fclsopn 24022	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
6	4, 2, 5	syl2anc 584	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
7	6	ibi 267	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
8		eleq2 2830	. . . . . 6 ⊢ (𝑜 = 𝑈 → (𝐴 ∈ 𝑜 ↔ 𝐴 ∈ 𝑈))
9		ineq1 4213	. . . . . . . 8 ⊢ (𝑜 = 𝑈 → (𝑜 ∩ 𝑠) = (𝑈 ∩ 𝑠))
10	9	neeq1d 3000	. . . . . . 7 ⊢ (𝑜 = 𝑈 → ((𝑜 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑠) ≠ ∅))
11	10	ralbidv 3178	. . . . . 6 ⊢ (𝑜 = 𝑈 → (∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅))
12	8, 11	imbi12d 344	. . . . 5 ⊢ (𝑜 = 𝑈 → ((𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) ↔ (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅)))
13	12	rspccv 3619	. . . 4 ⊢ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅)))
14	7, 13	simpl2im 503	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅)))
15		ineq2 4214	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑈 ∩ 𝑠) = (𝑈 ∩ 𝑆))
16	15	neeq1d 3000	. . . 4 ⊢ (𝑠 = 𝑆 → ((𝑈 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑆) ≠ ∅))
17	16	rspccv 3619	. . 3 ⊢ (∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅ → (𝑆 ∈ 𝐹 → (𝑈 ∩ 𝑆) ≠ ∅))
18	14, 17	syl8 76	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → (𝑆 ∈ 𝐹 → (𝑈 ∩ 𝑆) ≠ ∅))))
19	18	3imp2 1350	1 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061   ∩ cin 3950  ∅c0 4333  ∪ cuni 4907  ‘cfv 6561  (class class class)co 7431  TopOnctopon 22916  Filcfil 23853   fClus cfcls 23944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-top 22900  df-topon 22917  df-cld 23027  df-ntr 23028  df-cls 23029  df-fil 23854  df-fcls 23949

This theorem is referenced by:  fclsneii  24025  supnfcls  24028  flimfnfcls  24036  cfilfcls  25308