Theorem fclsopn 24043

Description: Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

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

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

Proof of Theorem fclsopn

Step	Hyp	Ref	Expression
1		isfcls2 24042	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
2		filn0 23891	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
3	2	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ≠ ∅)
4		r19.2z 4518	. . . . . 6 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
5	4	ex 412	. . . . 5 ⊢ (𝐹 ≠ ∅ → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
6	3, 5	syl 17	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
7		topontop 22940	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
8	7	ad2antrr 725	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝐽 ∈ Top)
9		filelss 23881	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋)
10	9	adantll 713	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋)
11		toponuni 22941	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
12	11	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑋 = ∪ 𝐽)
13	10, 12	sseqtrd 4049	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ ∪ 𝐽)
14		eqid 2740	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14	clsss3 23088	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑠) ⊆ ∪ 𝐽)
16	8, 13, 15	syl2anc 583	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → ((cls‘𝐽)‘𝑠) ⊆ ∪ 𝐽)
17	16, 12	sseqtrrd 4050	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝑋)
18	17	sseld 4007	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋))
19	18	rexlimdva 3161	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋))
20	6, 19	syld 47	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋))
21	20	pm4.71rd 562	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
22	7	ad3antrrr 729	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐽 ∈ Top)
23	13	adantlr 714	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ ∪ 𝐽)
24		simplr 768	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐴 ∈ 𝑋)
25	11	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑋 = ∪ 𝐽)
26	24, 25	eleqtrd 2846	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐴 ∈ ∪ 𝐽)
27	14	elcls 23102	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽 ∧ 𝐴 ∈ ∪ 𝐽) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅)))
28	22, 23, 26, 27	syl3anc 1371	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅)))
29	28	ralbidva 3182	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅)))
30		ralcom 3295	. . . . 5 ⊢ (∀𝑠 ∈ 𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 ∀𝑠 ∈ 𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅))
31		r19.21v 3186	. . . . . 6 ⊢ (∀𝑠 ∈ 𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
32	31	ralbii 3099	. . . . 5 ⊢ (∀𝑜 ∈ 𝐽 ∀𝑠 ∈ 𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
33	30, 32	bitri 275	. . . 4 ⊢ (∀𝑠 ∈ 𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
34	29, 33	bitrdi 287	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
35	34	pm5.32da 578	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
36	1, 21, 35	3bitrd 305	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ‘cfv 6573  (class class class)co 7448  Topctop 22920  TopOnctopon 22937  clsccl 23047  Filcfil 23874   fClus cfcls 23965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-fbas 21384  df-top 22921  df-topon 22938  df-cld 23048  df-ntr 23049  df-cls 23050  df-fil 23875  df-fcls 23970

This theorem is referenced by:  fclsopni  24044  fclselbas  24045  fclsnei  24048  fclsbas  24050  fclsss1  24051  fclsrest  24053  fclscf  24054  isfcf  24063