Theorem fclsbas 24345

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 24202	. . . . 5 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
3	2	adantl 487	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
4	1, 3	eqeltrid 2864	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
5		fclsopn 24338	. . 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 603	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅))))
7		ssfg 24196	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
8	7	ad3antlr 744	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐵 ⊆ (𝑋filGen𝐵))
9	8, 1	sseqtrrdi 3949	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐵 ⊆ 𝐹)
10		ssralv 3977	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐹 → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ → ∀𝑡 ∈ 𝐵 (𝑜 ∩ 𝑡) ≠ ∅))
11	9, 10	syl 18	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ → ∀𝑡 ∈ 𝐵 (𝑜 ∩ 𝑡) ≠ ∅))
12		ineq2 4111	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝑜 ∩ 𝑡) = (𝑜 ∩ 𝑠))
13	12	neeq1d 3014	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((𝑜 ∩ 𝑡) ≠ ∅ ↔ (𝑜 ∩ 𝑠) ≠ ∅))
14	13	cbvralvw 3240	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝐵 (𝑜 ∩ 𝑡) ≠ ∅ ↔ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅)
15	11, 14	imbitrdi 254	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))
16	1	eleq2i 2852	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝐹 ↔ 𝑡 ∈ (𝑋filGen𝐵))
17		elfg 24195	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
18	17	ad3antlr 744	. . . . . . . . . . 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 286	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (𝑡 ∈ 𝐹 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
20	19	simplbda 505	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) ∧ 𝑡 ∈ 𝐹) → ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)
21		r19.29r 3126	. . . . . . . . . . 11 ⊢ ((∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 ∧ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅) → ∃𝑠 ∈ 𝐵 (𝑠 ⊆ 𝑡 ∧ (𝑜 ∩ 𝑠) ≠ ∅))
22		sslin 4139	. . . . . . . . . . . . 13 ⊢ (𝑠 ⊆ 𝑡 → (𝑜 ∩ 𝑠) ⊆ (𝑜 ∩ 𝑡))
23		ssn0 4354	. . . . . . . . . . . . 13 ⊢ (((𝑜 ∩ 𝑠) ⊆ (𝑜 ∩ 𝑡) ∧ (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
24	22, 23	sylan 592	. . . . . . . . . . . 12 ⊢ ((𝑠 ⊆ 𝑡 ∧ (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
25	24	rexlimivw 3159	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ 𝐵 (𝑠 ⊆ 𝑡 ∧ (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
26	21, 25	syl 18	. . . . . . . . . 10 ⊢ ((∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 ∧ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
27	26	ex 418	. . . . . . . . 9 ⊢ (∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 → (∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅ → (𝑜 ∩ 𝑡) ≠ ∅))
28	20, 27	syl 18	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) ∧ 𝑡 ∈ 𝐹) → (∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅ → (𝑜 ∩ 𝑡) ≠ ∅))
29	28	ralrimdva 3162	. . . . . . 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 215	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ ↔ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))
31	30	anassrs 473	. . . . 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 816	. . . 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 3183	. . 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 590	. 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 282	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  ‘cfv 6538  (class class class)co 7420  fBascfbas 21671  filGencfg 21672  TopOnctopon 23233  Filcfil 24169   fClus cfcls 24260

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-fbas 21680  df-fg 21681  df-top 23217  df-topon 23234  df-cld 23342  df-ntr 23343  df-cls 23344  df-fil 24170  df-fcls 24265

This theorem is used by: (None)