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Theorem fclsrest 24032

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 1137	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2		filelss 23860	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
3	2	3adant1 1131	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
4		resttopon 23169	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
5	1, 3, 4	syl2anc 584	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
6		filfbas 23856	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
7	6	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
8		simp3 1139	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ∈ 𝐹)
9		fbncp 23847	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
10	7, 8, 9	syl2anc 584	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
11		simp2 1138	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12		trfil3 23896	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝐹 ↾_t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
13	11, 3, 12	syl2anc 584	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐹 ↾_t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
14	10, 13	mpbird 257	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ↾_t 𝑌) ∈ (Fil‘𝑌))
15		fclsopn 24022	. . . . 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 584	. . . 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 4230	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ 𝑠)
18		ineq2 4214	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → ((𝑢 ∩ 𝑌) ∩ 𝑠) = ((𝑢 ∩ 𝑌) ∩ 𝑡))
19	17, 18	eqtrid 2789	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ 𝑡))
20	19	neeq1d 3000	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅))
21	20	rspccv 3619	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → (𝑡 ∈ 𝐹 → ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅))
22		inss1 4237	. . . . . . . . . . . . 13 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑢
23		ssrin 4242	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝑌) ⊆ 𝑢 → ((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡))
24	22, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡)
25		ssn0 4404	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡) ∧ ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅) → (𝑢 ∩ 𝑡) ≠ ∅)
26	24, 25	mpan 690	. . . . . . . . . . 11 ⊢ (((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅ → (𝑢 ∩ 𝑡) ≠ ∅)
27	21, 26	syl6 35	. . . . . . . . . 10 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → (𝑡 ∈ 𝐹 → (𝑢 ∩ 𝑡) ≠ ∅))
28	27	ralrimiv 3145	. . . . . . . . 9 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅)
29	11	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
30		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝑠 ∈ 𝐹)
31	8	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝑌 ∈ 𝐹)
32		filin 23862	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ 𝐹)
33	29, 30, 31, 32	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ 𝐹)
34		ineq2 4214	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → (𝑢 ∩ 𝑡) = (𝑢 ∩ (𝑠 ∩ 𝑌)))
35		inass 4228	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠 ∩ 𝑌))
36	34, 35	eqtr4di 2795	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → (𝑢 ∩ 𝑡) = ((𝑢 ∩ 𝑠) ∩ 𝑌))
37	36	neeq1d 3000	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → ((𝑢 ∩ 𝑡) ≠ ∅ ↔ ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
38	37	rspcv 3618	. . . . . . . . . . 11 ⊢ ((𝑠 ∩ 𝑌) ∈ 𝐹 → (∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅ → ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
39	33, 38	syl 17	. . . . . . . . . 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 3154	. . . . . . . . 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 226	. . . . . . . 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 340	. . . . . . 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 3176	. . . . . 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 3484	. . . . . . . . 9 ⊢ 𝑢 ∈ V
45	44	inex1 5317	. . . . . . . 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 17472	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
48	47	3adant2 1132	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
49	48	adantr 480	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
50		eleq2 2830	. . . . . . . . 9 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑢 ∩ 𝑌)))
51		elin 3967	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑌))
52	51	rbaib 538	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 → (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ 𝑥 ∈ 𝑢))
53	52	adantl 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ 𝑥 ∈ 𝑢))
54	50, 53	sylan9bbr 510	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑢 ∩ 𝑌)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑢))
55		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
56	55	inex1 5317	. . . . . . . . . . 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 17472	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
59	58	3adant1 1131	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
60	59	adantr 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
61		ineq2 4214	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑠 ∩ 𝑌) → (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑠 ∩ 𝑌)))
62	61	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 = (𝑠 ∩ 𝑌)) → (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑠 ∩ 𝑌)))
63	62	neeq1d 3000	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 = (𝑠 ∩ 𝑌)) → ((𝑦 ∩ 𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅))
64	57, 60, 63	ralxfr2d 5410	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅))
65		ineq1 4213	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑦 ∩ (𝑠 ∩ 𝑌)) = ((𝑢 ∩ 𝑌) ∩ (𝑠 ∩ 𝑌)))
66		inindir 4236	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ (𝑠 ∩ 𝑌))
67	65, 66	eqtr4di 2795	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑦 ∩ (𝑠 ∩ 𝑌)) = ((𝑢 ∩ 𝑠) ∩ 𝑌))
68	67	neeq1d 3000	. . . . . . . . . 10 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → ((𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅ ↔ ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
69	68	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (∀𝑠 ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
70	64, 69	sylan9bb 509	. . . . . . . 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 344	. . . . . . 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 5410	. . . . . 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 480	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
74	11	adantr 480	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐹 ∈ (Fil‘𝑋))
75	3	sselda 3983	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
76		fclsopn 24022	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅))))
77	76	baibd 539	. . . . . . 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 838	. . . . . 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 311	. . . . 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 579	. . . 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 279	. . 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 3967	. . . 4 ⊢ (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fClus 𝐹) ∧ 𝑥 ∈ 𝑌))
83	82	biancomi 462	. . 3 ⊢ (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fClus 𝐹)))
84	81, 83	bitr4di 289	. 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 2735	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 ↾_t crest 17465 fBascfbas 21352 TopOnctopon 22916 Filcfil 23853 fClus cfcls 23944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-fbas 21361 df-fg 21362 df-top 22900 df-topon 22917 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-fil 23854 df-fcls 23949

This theorem is referenced by: relcmpcmet 25352

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