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Theorem fclsrest 24142

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 1152	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2		filelss 23970	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
3	2	3adant1 1146	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
4		resttopon 23279	. . . . . 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 23966	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
7	6	3ad2ant2 1150	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
8		simp3 1154	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ∈ 𝐹)
9		fbncp 23957	. . . . . . 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 1153	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12		trfil3 24006	. . . . . . 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 24132	. . . . 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 4184	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ 𝑠)
18		ineq2 4169	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → ((𝑢 ∩ 𝑌) ∩ 𝑠) = ((𝑢 ∩ 𝑌) ∩ 𝑡))
19	17, 18	eqtrid 2812	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ 𝑡))
20	19	neeq1d 3019	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅))
21	20	rspccv 3581	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → (𝑡 ∈ 𝐹 → ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅))
22		inss1 4191	. . . . . . . . . . . . 13 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑢
23		ssrin 4196	. . . . . . . . . . . . 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 3156	. . . . . . . . 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 23972	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ 𝐹)
33	29, 30, 31, 32	syl3anc 1394	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ 𝐹)
34		ineq2 4169	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → (𝑢 ∩ 𝑡) = (𝑢 ∩ (𝑠 ∩ 𝑌)))
35		inass 4182	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠 ∩ 𝑌))
36	34, 35	eqtr4di 2818	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → (𝑢 ∩ 𝑡) = ((𝑢 ∩ 𝑠) ∩ 𝑌))
37	36	neeq1d 3019	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → ((𝑢 ∩ 𝑡) ≠ ∅ ↔ ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
38	37	rspcv 3580	. . . . . . . . . . 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 3165	. . . . . . . . 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 3186	. . . . . 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 3461	. . . . . . . . 9 ⊢ 𝑢 ∈ V
45	44	inex1 5278	. . . . . . . 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 17470	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
48	47	3adant2 1147	. . . . . . . 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 2854	. . . . . . . . 9 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑢 ∩ 𝑌)))
51		elin 3923	. . . . . . . . . . 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 3461	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
56	55	inex1 5278	. . . . . . . . . . 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 17470	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
59	58	3adant1 1146	. . . . . . . . . . 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 4169	. . . . . . . . . . . 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 3019	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 = (𝑠 ∩ 𝑌)) → ((𝑦 ∩ 𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅))
64	57, 60, 63	ralxfr2d 5372	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅))
65		ineq1 4168	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑦 ∩ (𝑠 ∩ 𝑌)) = ((𝑢 ∩ 𝑌) ∩ (𝑠 ∩ 𝑌)))
66		inindir 4190	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ (𝑠 ∩ 𝑌))
67	65, 66	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑦 ∩ (𝑠 ∩ 𝑌)) = ((𝑢 ∩ 𝑠) ∩ 𝑌))
68	67	neeq1d 3019	. . . . . . . . . 10 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → ((𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅ ↔ ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
69	68	ralbidv 3188	. . . . . . . . 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 5372	. . . . . 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 3939	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
76		fclsopn 24132	. . . . . . . 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 3923	. . . 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 2763	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 Vcvv 3457 ∖ cdif 3904 ∩ cin 3906 ⊆ wss 3907 ∅c0 4288 ‘cfv 6525 (class class class)co 7400 ↾_t crest 17463 fBascfbas 21470 TopOnctopon 23028 Filcfil 23963 fClus cfcls 24054

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-en 8932 df-fin 8935 df-fi 9359 df-rest 17465 df-topgen 17486 df-fbas 21479 df-fg 21480 df-top 23012 df-topon 23029 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-fil 23964 df-fcls 24059

This theorem is referenced by: relcmpcmet 25438

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