Theorem fclsopni 22327

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 2779	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclsfil 22322	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
3		fclstopon 22324	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ 𝐹 ∈ (Fil‘∪ 𝐽)))
4	2, 3	mpbird 249	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		fclsopn 22326	. . . . . 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 576	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
7	6	ibi 259	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
8		eleq2 2855	. . . . . 6 ⊢ (𝑜 = 𝑈 → (𝐴 ∈ 𝑜 ↔ 𝐴 ∈ 𝑈))
9		ineq1 4069	. . . . . . . 8 ⊢ (𝑜 = 𝑈 → (𝑜 ∩ 𝑠) = (𝑈 ∩ 𝑠))
10	9	neeq1d 3027	. . . . . . 7 ⊢ (𝑜 = 𝑈 → ((𝑜 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑠) ≠ ∅))
11	10	ralbidv 3148	. . . . . 6 ⊢ (𝑜 = 𝑈 → (∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅))
12	8, 11	imbi12d 337	. . . . 5 ⊢ (𝑜 = 𝑈 → ((𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) ↔ (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅)))
13	12	rspccv 3533	. . . 4 ⊢ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅)))
14	7, 13	simpl2im 496	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅)))
15		ineq2 4071	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑈 ∩ 𝑠) = (𝑈 ∩ 𝑆))
16	15	neeq1d 3027	. . . 4 ⊢ (𝑠 = 𝑆 → ((𝑈 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑆) ≠ ∅))
17	16	rspccv 3533	. . 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 1329	1 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2968  ∀wral 3089   ∩ cin 3829  ∅c0 4179  ∪ cuni 4712  ‘cfv 6188  (class class class)co 6976  TopOnctopon 21222  Filcfil 22157   fClus cfcls 22248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-fbas 20244  df-top 21206  df-topon 21223  df-cld 21331  df-ntr 21332  df-cls 21333  df-fil 22158  df-fcls 22253

This theorem is referenced by:  fclsneii  22329  supnfcls  22332  flimfnfcls  22340  cfilfcls  23580