Theorem fclsnei 24371

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 2763	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclselbas 24368	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ ∪ 𝐽)
3		toponuni 23265	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
4	3	adantr 486	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝑋 = ∪ 𝐽)
5	4	eleq2d 2850	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ 𝐽))
6	2, 5	imbitrrid 249	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋))
7		fclsneii 24369	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑠 ∈ 𝐹) → (𝑛 ∩ 𝑠) ≠ ∅)
8	7	3expb 1138	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑠 ∈ 𝐹)) → (𝑛 ∩ 𝑠) ≠ ∅)
9	8	ralrimivva 3209	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)
10	6, 9	jca2 523	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)))
11		topontop 23264	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
12	11	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐽 ∈ Top)
13		simprl 783	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ 𝐽)
14		simprr 785	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐴 ∈ 𝑜)
15		opnneip 23470	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
16	12, 13, 14, 15	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
17		ineq1 4113	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑜 → (𝑛 ∩ 𝑠) = (𝑜 ∩ 𝑠))
18	17	neeq1d 3018	. . . . . . . . . 10 ⊢ (𝑛 = 𝑜 → ((𝑛 ∩ 𝑠) ≠ ∅ ↔ (𝑜 ∩ 𝑠) ≠ ∅))
19	18	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑛 = 𝑜 → (∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
20	19	rspcv 3576	. . . . . . . 8 ⊢ (𝑜 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅ → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
21	16, 20	syl 18	. . . . . . 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 462	. . . . . 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 87	. . . . 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 3166	. . . 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 582	. . 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 24366	. . 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 262	. 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 215	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ∩ cin 4092  ∅c0 4281  {csn 4588  ∪ cuni 4870  ‘cfv 6561  (class class class)co 7446  Topctop 23244  TopOnctopon 23261  neicnei 23448  Filcfil 24197   fClus cfcls 24288

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-fbas 21708  df-top 23245  df-topon 23262  df-cld 23370  df-ntr 23371  df-cls 23372  df-nei 23449  df-fil 24198  df-fcls 24293

This theorem is used by: (None)