Theorem fclsbas 23965

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 23822	. . . . 5 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
3	2	adantl 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
4	1, 3	eqeltrid 2840	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
5		fclsopn 23958	. . 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 23816	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
8	7	ad3antlr 731	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐵 ⊆ (𝑋filGen𝐵))
9	8, 1	sseqtrrdi 3975	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝐵 ⊆ 𝐹)
10		ssralv 4002	. . . . . . . . 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 4166	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝑜 ∩ 𝑡) = (𝑜 ∩ 𝑠))
13	12	neeq1d 2991	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((𝑜 ∩ 𝑡) ≠ ∅ ↔ (𝑜 ∩ 𝑠) ≠ ∅))
14	13	cbvralvw 3214	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝐵 (𝑜 ∩ 𝑡) ≠ ∅ ↔ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅)
15	11, 14	imbitrdi 251	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))
16	1	eleq2i 2828	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝐹 ↔ 𝑡 ∈ (𝑋filGen𝐵))
17		elfg 23815	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
18	17	ad3antlr 731	. . . . . . . . . . 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 283	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (𝑡 ∈ 𝐹 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
20	19	simplbda 499	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) ∧ 𝑡 ∈ 𝐹) → ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)
21		r19.29r 3100	. . . . . . . . . . 11 ⊢ ((∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 ∧ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅) → ∃𝑠 ∈ 𝐵 (𝑠 ⊆ 𝑡 ∧ (𝑜 ∩ 𝑠) ≠ ∅))
22		sslin 4195	. . . . . . . . . . . . 13 ⊢ (𝑠 ⊆ 𝑡 → (𝑜 ∩ 𝑠) ⊆ (𝑜 ∩ 𝑡))
23		ssn0 4356	. . . . . . . . . . . . 13 ⊢ (((𝑜 ∩ 𝑠) ⊆ (𝑜 ∩ 𝑡) ∧ (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
24	22, 23	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝑠 ⊆ 𝑡 ∧ (𝑜 ∩ 𝑠) ≠ ∅) → (𝑜 ∩ 𝑡) ≠ ∅)
25	24	rexlimivw 3133	. . . . . . . . . . 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 412	. . . . . . . . 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 3136	. . . . . . 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 212	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑡 ∈ 𝐹 (𝑜 ∩ 𝑡) ≠ ∅ ↔ ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))
31	30	anassrs 467	. . . . 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 803	. . . 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 3157	. . 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 279	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  ‘cfv 6492  (class class class)co 7358  fBascfbas 21297  filGencfg 21298  TopOnctopon 22854  Filcfil 23789   fClus cfcls 23880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  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 7361  df-oprab 7362  df-mpo 7363  df-fbas 21306  df-fg 21307  df-top 22838  df-topon 22855  df-cld 22963  df-ntr 22964  df-cls 22965  df-fil 23790  df-fcls 23885

This theorem is referenced by: (None)