Theorem fclsnei 23994

Description: Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclsnei	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)))

Distinct variable groups:   𝑛,𝑠,𝐴   𝑛,𝐹,𝑠   𝑛,𝐽,𝑠   𝑋,𝑠

Allowed substitution hint:   𝑋(𝑛)

Proof of Theorem fclsnei

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclselbas 23991	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ ∪ 𝐽)
3		toponuni 22889	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
4	3	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝑋 = ∪ 𝐽)
5	4	eleq2d 2823	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ 𝐽))
6	2, 5	imbitrrid 246	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋))
7		fclsneii 23992	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑠 ∈ 𝐹) → (𝑛 ∩ 𝑠) ≠ ∅)
8	7	3expb 1121	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑠 ∈ 𝐹)) → (𝑛 ∩ 𝑠) ≠ ∅)
9	8	ralrimivva 3181	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)
10	6, 9	jca2 513	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)))
11		topontop 22888	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
12	11	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐽 ∈ Top)
13		simprl 771	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ 𝐽)
14		simprr 773	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐴 ∈ 𝑜)
15		opnneip 23094	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
16	12, 13, 14, 15	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
17		ineq1 4154	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑜 → (𝑛 ∩ 𝑠) = (𝑜 ∩ 𝑠))
18	17	neeq1d 2992	. . . . . . . . . 10 ⊢ (𝑛 = 𝑜 → ((𝑛 ∩ 𝑠) ≠ ∅ ↔ (𝑜 ∩ 𝑠) ≠ ∅))
19	18	ralbidv 3161	. . . . . . . . 9 ⊢ (𝑛 = 𝑜 → (∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
20	19	rspcv 3561	. . . . . . . 8 ⊢ (𝑜 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅ → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
21	16, 20	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅ → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
22	21	expr 456	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐴 ∈ 𝑜 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅ → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
23	22	com23 86	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅ → (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
24	23	ralrimdva 3138	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅ → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
25	24	imdistanda 571	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅) → (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
26		fclsopn 23989	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
27	25, 26	sylibrd 259	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅) → 𝐴 ∈ (𝐽 fClus 𝐹)))
28	10, 27	impbid 212	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∩ cin 3889  ∅c0 4274  {csn 4568  ∪ cuni 4851  ‘cfv 6492  (class class class)co 7360  Topctop 22868  TopOnctopon 22885  neicnei 23072  Filcfil 23820   fClus cfcls 23911

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-fbas 21341  df-top 22869  df-topon 22886  df-cld 22994  df-ntr 22995  df-cls 22996  df-nei 23073  df-fil 23821  df-fcls 23916

This theorem is referenced by: (None)