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Theorem fclsrest 24192

Description: The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)

**Assertion**
Ref	Expression
fclsrest	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾_t 𝑌) fClus (𝐹 ↾_t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌))

Proof of Theorem fclsrest Dummy variables 𝑠 𝑡 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1153	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2		filelss 24020	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
3	2	3adant1 1147	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
4		resttopon 23329	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
5	1, 3, 4	syl2anc 595	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
6		filfbas 24016	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
7	6	3ad2ant2 1151	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
8		simp3 1155	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ∈ 𝐹)
9		fbncp 24007	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
10	7, 8, 9	syl2anc 595	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
11		simp2 1154	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12		trfil3 24056	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝐹 ↾_t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
13	11, 3, 12	syl2anc 595	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐹 ↾_t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
14	10, 13	mpbird 260	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ↾_t 𝑌) ∈ (Fil‘𝑌))
15		fclsopn 24182	. . . . 5 ⊢ (((𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌) ∧ (𝐹 ↾_t 𝑌) ∈ (Fil‘𝑌)) → (𝑥 ∈ ((𝐽 ↾_t 𝑌) fClus (𝐹 ↾_t 𝑌)) ↔ (𝑥 ∈ 𝑌 ∧ ∀𝑦 ∈ (𝐽 ↾_t 𝑌)(𝑥 ∈ 𝑦 → ∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅))))
16	5, 14, 15	syl2anc 595	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑥 ∈ ((𝐽 ↾_t 𝑌) fClus (𝐹 ↾_t 𝑌)) ↔ (𝑥 ∈ 𝑌 ∧ ∀𝑦 ∈ (𝐽 ↾_t 𝑌)(𝑥 ∈ 𝑦 → ∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅))))
17		in32 4181	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ 𝑠)
18		ineq2 4166	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → ((𝑢 ∩ 𝑌) ∩ 𝑠) = ((𝑢 ∩ 𝑌) ∩ 𝑡))
19	17, 18	eqtrid 2809	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ 𝑡))
20	19	neeq1d 3016	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅))
21	20	rspccv 3577	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → (𝑡 ∈ 𝐹 → ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅))
22		inss1 4188	. . . . . . . . . . . . 13 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑢
23		ssrin 4193	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝑌) ⊆ 𝑢 → ((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡))
24	22, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡)
25		ssn0 4361	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡) ∧ ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅) → (𝑢 ∩ 𝑡) ≠ ∅)
26	24, 25	mpan 702	. . . . . . . . . . 11 ⊢ (((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅ → (𝑢 ∩ 𝑡) ≠ ∅)
27	21, 26	syl6 36	. . . . . . . . . 10 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → (𝑡 ∈ 𝐹 → (𝑢 ∩ 𝑡) ≠ ∅))
28	27	ralrimiv 3155	. . . . . . . . 9 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅)
29	11	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
30		simpr 489	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝑠 ∈ 𝐹)
31	8	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝑌 ∈ 𝐹)
32		filin 24022	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ 𝐹)
33	29, 30, 31, 32	syl3anc 1397	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ 𝐹)
34		ineq2 4166	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → (𝑢 ∩ 𝑡) = (𝑢 ∩ (𝑠 ∩ 𝑌)))
35		inass 4179	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠 ∩ 𝑌))
36	34, 35	eqtr4di 2815	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → (𝑢 ∩ 𝑡) = ((𝑢 ∩ 𝑠) ∩ 𝑌))
37	36	neeq1d 3016	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → ((𝑢 ∩ 𝑡) ≠ ∅ ↔ ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
38	37	rspcv 3576	. . . . . . . . . . 11 ⊢ ((𝑠 ∩ 𝑌) ∈ 𝐹 → (∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅ → ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
39	33, 38	syl 18	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → (∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅ → ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
40	39	ralrimdva 3164	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) → (∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅ → ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
41	28, 40	impbid2 229	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) → (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ ↔ ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅))
42	41	imbi2d 343	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) → ((𝑥 ∈ 𝑢 → ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅) ↔ (𝑥 ∈ 𝑢 → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅)))
43	42	ralbidva 3185	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅) ↔ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅)))
44		vex 3458	. . . . . . . . 9 ⊢ 𝑢 ∈ V
45	44	inex1 5285	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑌) ∈ V
46	45	a1i 11	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) → (𝑢 ∩ 𝑌) ∈ V)
47		elrest 17486	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
48	47	3adant2 1148	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
49	48	adantr 485	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
50		eleq2 2851	. . . . . . . . 9 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑢 ∩ 𝑌)))
51		elin 3920	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑌))
52	51	rbaib 547	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 → (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ 𝑥 ∈ 𝑢))
53	52	adantl 486	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ 𝑥 ∈ 𝑢))
54	50, 53	sylan9bbr 519	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑢 ∩ 𝑌)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑢))
55		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
56	55	inex1 5285	. . . . . . . . . . 11 ⊢ (𝑠 ∩ 𝑌) ∈ V
57	56	a1i 11	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑠 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ V)
58		elrest 17486	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
59	58	3adant1 1147	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
60	59	adantr 485	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
61		ineq2 4166	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑠 ∩ 𝑌) → (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑠 ∩ 𝑌)))
62	61	adantl 486	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 = (𝑠 ∩ 𝑌)) → (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑠 ∩ 𝑌)))
63	62	neeq1d 3016	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 = (𝑠 ∩ 𝑌)) → ((𝑦 ∩ 𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅))
64	57, 60, 63	ralxfr2d 5380	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅))
65		ineq1 4165	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑦 ∩ (𝑠 ∩ 𝑌)) = ((𝑢 ∩ 𝑌) ∩ (𝑠 ∩ 𝑌)))
66		inindir 4187	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ (𝑠 ∩ 𝑌))
67	65, 66	eqtr4di 2815	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑦 ∩ (𝑠 ∩ 𝑌)) = ((𝑢 ∩ 𝑠) ∩ 𝑌))
68	67	neeq1d 3016	. . . . . . . . . 10 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → ((𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅ ↔ ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
69	68	ralbidv 3187	. . . . . . . . 9 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (∀𝑠 ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
70	64, 69	sylan9bb 518	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑢 ∩ 𝑌)) → (∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
71	54, 70	imbi12d 347	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑢 ∩ 𝑌)) → ((𝑥 ∈ 𝑦 → ∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅) ↔ (𝑥 ∈ 𝑢 → ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅)))
72	46, 49, 71	ralxfr2d 5380	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑦 ∈ (𝐽 ↾_t 𝑌)(𝑥 ∈ 𝑦 → ∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅) ↔ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅)))
73	1	adantr 485	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
74	11	adantr 485	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐹 ∈ (Fil‘𝑋))
75	3	sselda 3936	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
76		fclsopn 24182	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅))))
77	76	baibd 548	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅)))
78	73, 74, 75, 77	syl21anc 850	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅)))
79	43, 72, 78	3bitr4d 314	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑦 ∈ (𝐽 ↾_t 𝑌)(𝑥 ∈ 𝑦 → ∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅) ↔ 𝑥 ∈ (𝐽 fClus 𝐹)))
80	79	pm5.32da 589	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝑥 ∈ 𝑌 ∧ ∀𝑦 ∈ (𝐽 ↾_t 𝑌)(𝑥 ∈ 𝑦 → ∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅)) ↔ (𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fClus 𝐹))))
81	16, 80	bitrd 282	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑥 ∈ ((𝐽 ↾_t 𝑌) fClus (𝐹 ↾_t 𝑌)) ↔ (𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fClus 𝐹))))
82		elin 3920	. . . 4 ⊢ (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fClus 𝐹) ∧ 𝑥 ∈ 𝑌))
83	82	biancomi 467	. . 3 ⊢ (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fClus 𝐹)))
84	81, 83	bitr4di 292	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑥 ∈ ((𝐽 ↾_t 𝑌) fClus (𝐹 ↾_t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌)))
85	84	eqrdv 2760	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾_t 𝑌) fClus (𝐹 ↾_t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 Vcvv 3454 ∖ cdif 3901 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 ‘cfv 6536 (class class class)co 7412 ↾_t crest 17479 fBascfbas 21521 TopOnctopon 23078 Filcfil 24013 fClus cfcls 24104

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-en 8942 df-fin 8945 df-fi 9369 df-rest 17481 df-topgen 17502 df-fbas 21530 df-fg 21531 df-top 23062 df-topon 23079 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-fil 24014 df-fcls 24109

This theorem is used by: relcmpcmet 25488

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