Theorem fclsopni 23971

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 23966	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
3		fclstopon 23968	. . . . . . 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 23970	. . . . . 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 585	. . . . 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 2826	. . . . . 6 ⊢ (𝑜 = 𝑈 → (𝐴 ∈ 𝑜 ↔ 𝐴 ∈ 𝑈))
9		ineq1 4167	. . . . . . . 8 ⊢ (𝑜 = 𝑈 → (𝑜 ∩ 𝑠) = (𝑈 ∩ 𝑠))
10	9	neeq1d 2992	. . . . . . 7 ⊢ (𝑜 = 𝑈 → ((𝑜 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑠) ≠ ∅))
11	10	ralbidv 3161	. . . . . 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 3575	. . . 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 4168	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑈 ∩ 𝑠) = (𝑈 ∩ 𝑆))
16	15	neeq1d 2992	. . . 4 ⊢ (𝑠 = 𝑆 → ((𝑈 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑆) ≠ ∅))
17	16	rspccv 3575	. . 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 1351	1 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∩ cin 3902  ∅c0 4287  ∪ cuni 4865  ‘cfv 6500  (class class class)co 7368  TopOnctopon 22866  Filcfil 23801   fClus cfcls 23892

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-fbas 21318  df-top 22850  df-topon 22867  df-cld 22975  df-ntr 22976  df-cls 22977  df-fil 23802  df-fcls 23897

This theorem is referenced by:  fclsneii  23973  supnfcls  23976  flimfnfcls  23984  cfilfcls  25242