Theorem fclsbas 23532

Description: Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypothesis

Ref	Expression
fclsbas.f	⊢ 𝐹 = (𝑋filGen𝐵)

Assertion

Ref	Expression
fclsbas	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))))

Distinct variable groups:   𝐴,𝑜   𝑜,𝑠,𝐵   𝑜,𝐹   𝑜,𝐽   𝑜,𝑋

Allowed substitution hints:   𝐴(𝑠)   𝐹(𝑠)   𝐽(𝑠)   𝑋(𝑠)

Proof of Theorem fclsbas

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fclsbas.f	. . . 4 ⊢ 𝐹 = (𝑋filGen𝐵)
2		fgcl 23389	. . . . 5 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
3	2	adantl 482	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
4	1, 3	eqeltrid 2837	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
5		fclsopn 23525	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅))))
6	4, 5	syldan 591	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅))))
7		ssfg 23383	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
8	7	ad3antlr 729	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐵 ⊆ (𝑋filGen𝐵))
9	8, 1	sseqtrrdi 4033	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐵 ⊆ 𝐹)
10		ssralv 4050	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐹 → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ → ∀𝑡 ∈ 𝐵 (𝑜 ∩ 𝑡) ≠ ∅))
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ → ∀𝑡 ∈ 𝐵 (𝑜 ∩ 𝑡) ≠ ∅))
12		ineq2 4206	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝑜 ∩ 𝑡) = (𝑜 ∩ 𝑠))
13	12	neeq1d 3000	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((𝑜 ∩ 𝑡) ≠ ∅ ↔ (𝑜 ∩ 𝑠) ≠ ∅))
14	13	cbvralvw 3234	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝐵 (𝑜 ∩ 𝑡) ≠ ∅ ↔ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅)
15	11, 14	imbitrdi 250	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))
16	1	eleq2i 2825	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝐹 ↔ 𝑡 ∈ (𝑋filGen𝐵))
17		elfg 23382	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
18	17	ad3antlr 729	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
19	16, 18	bitrid 282	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (𝑡 ∈ 𝐹 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
20	19	simplbda 500	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) ∧ 𝑡 ∈ 𝐹) → ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)
21		r19.29r 3116	. . . . . . . . . . 11 ⊢ ((∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 ∧ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅) → ∃𝑠 ∈ 𝐵 (𝑠 ⊆ 𝑡 ∧ (𝑜 ∩ 𝑠) ≠ ∅))
22		sslin 4234	. . . . . . . . . . . . 13 ⊢ (𝑠 ⊆ 𝑡 → (𝑜 ∩ 𝑠) ⊆ (𝑜 ∩ 𝑡))
23		ssn0 4400	. . . . . . . . . . . . 13 ⊢ (((𝑜 ∩ 𝑠) ⊆ (𝑜 ∩ 𝑡) ∧ (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
24	22, 23	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝑠 ⊆ 𝑡 ∧ (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
25	24	rexlimivw 3151	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ 𝐵 (𝑠 ⊆ 𝑡 ∧ (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
26	21, 25	syl 17	. . . . . . . . . 10 ⊢ ((∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 ∧ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
27	26	ex 413	. . . . . . . . 9 ⊢ (∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 → (∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅ → (𝑜 ∩ 𝑡) ≠ ∅))
28	20, 27	syl 17	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) ∧ 𝑡 ∈ 𝐹) → (∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅ → (𝑜 ∩ 𝑡) ≠ ∅))
29	28	ralrimdva 3154	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅ → ∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅))
30	15, 29	impbid 211	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ ↔ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))
31	30	anassrs 468	. . . . 5 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝐴 ∈ 𝑜) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ ↔ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))
32	31	pm5.74da 802	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → ((𝐴 ∈ 𝑜 → ∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅) ↔ (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅)))
33	32	ralbidva 3175	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅)))
34	33	pm5.32da 579	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))))
35	6, 34	bitrd 278	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  ‘cfv 6543  (class class class)co 7411  fBascfbas 20938  filGencfg 20939  TopOnctopon 22419  Filcfil 23356   fClus cfcls 23447

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 20947  df-fg 20948  df-top 22403  df-topon 22420  df-cld 22530  df-ntr 22531  df-cls 22532  df-fil 23357  df-fcls 23452

This theorem is referenced by: (None)