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Theorem fclsrest 23175

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 1135	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2		filelss 23003	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
3	2	3adant1 1129	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
4		resttopon 22312	. . . . . 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 22999	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
7	6	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
8		simp3 1137	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ∈ 𝐹)
9		fbncp 22990	. . . . . . 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 1136	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12		trfil3 23039	. . . . . . 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 256	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ↾_t 𝑌) ∈ (Fil‘𝑌))
15		fclsopn 23165	. . . . 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 4155	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ 𝑠)
18		ineq2 4140	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → ((𝑢 ∩ 𝑌) ∩ 𝑠) = ((𝑢 ∩ 𝑌) ∩ 𝑡))
19	17, 18	eqtrid 2790	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ 𝑡))
20	19	neeq1d 3003	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅))
21	20	rspccv 3558	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → (𝑡 ∈ 𝐹 → ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅))
22		inss1 4162	. . . . . . . . . . . . 13 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑢
23		ssrin 4167	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝑌) ⊆ 𝑢 → ((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡))
24	22, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡)
25		ssn0 4334	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∩ 𝑌) ∩ 𝑡) ⊆ (𝑢 ∩ 𝑡) ∧ ((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅) → (𝑢 ∩ 𝑡) ≠ ∅)
26	24, 25	mpan 687	. . . . . . . . . . 11 ⊢ (((𝑢 ∩ 𝑌) ∩ 𝑡) ≠ ∅ → (𝑢 ∩ 𝑡) ≠ ∅)
27	21, 26	syl6 35	. . . . . . . . . 10 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → (𝑡 ∈ 𝐹 → (𝑢 ∩ 𝑡) ≠ ∅))
28	27	ralrimiv 3102	. . . . . . . . 9 ⊢ (∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅ → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅)
29	11	ad3antrrr 727	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
30		simpr 485	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝑠 ∈ 𝐹)
31	8	ad3antrrr 727	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → 𝑌 ∈ 𝐹)
32		filin 23005	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ 𝐹)
33	29, 30, 31, 32	syl3anc 1370	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) → (𝑠 ∩ 𝑌) ∈ 𝐹)
34		ineq2 4140	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → (𝑢 ∩ 𝑡) = (𝑢 ∩ (𝑠 ∩ 𝑌)))
35		inass 4153	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠 ∩ 𝑌))
36	34, 35	eqtr4di 2796	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → (𝑢 ∩ 𝑡) = ((𝑢 ∩ 𝑠) ∩ 𝑌))
37	36	neeq1d 3003	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑠 ∩ 𝑌) → ((𝑢 ∩ 𝑡) ≠ ∅ ↔ ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
38	37	rspcv 3557	. . . . . . . . . . 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 3106	. . . . . . . . 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 225	. . . . . . . 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 341	. . . . . . 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 3111	. . . . . 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 3436	. . . . . . . . 9 ⊢ 𝑢 ∈ V
45	44	inex1 5241	. . . . . . . 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 17138	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
48	47	3adant2 1130	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
49	48	adantr 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑦 ∈ (𝐽 ↾_t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑦 = (𝑢 ∩ 𝑌)))
50		eleq2 2827	. . . . . . . . 9 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑢 ∩ 𝑌)))
51		elin 3903	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑌))
52	51	rbaib 539	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 → (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ 𝑥 ∈ 𝑢))
53	52	adantl 482	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝑢 ∩ 𝑌) ↔ 𝑥 ∈ 𝑢))
54	50, 53	sylan9bbr 511	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑢 ∩ 𝑌)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑢))
55		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
56	55	inex1 5241	. . . . . . . . . . 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 17138	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
59	58	3adant1 1129	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
60	59	adantr 481	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑧 ∈ (𝐹 ↾_t 𝑌) ↔ ∃𝑠 ∈ 𝐹 𝑧 = (𝑠 ∩ 𝑌)))
61		ineq2 4140	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑠 ∩ 𝑌) → (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑠 ∩ 𝑌)))
62	61	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 = (𝑠 ∩ 𝑌)) → (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑠 ∩ 𝑌)))
63	62	neeq1d 3003	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 = (𝑠 ∩ 𝑌)) → ((𝑦 ∩ 𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅))
64	57, 60, 63	ralxfr2d 5333	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑧 ∈ (𝐹 ↾_t 𝑌)(𝑦 ∩ 𝑧) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅))
65		ineq1 4139	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑦 ∩ (𝑠 ∩ 𝑌)) = ((𝑢 ∩ 𝑌) ∩ (𝑠 ∩ 𝑌)))
66		inindir 4161	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑠) ∩ 𝑌) = ((𝑢 ∩ 𝑌) ∩ (𝑠 ∩ 𝑌))
67	65, 66	eqtr4di 2796	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (𝑦 ∩ (𝑠 ∩ 𝑌)) = ((𝑢 ∩ 𝑠) ∩ 𝑌))
68	67	neeq1d 3003	. . . . . . . . . 10 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → ((𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅ ↔ ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
69	68	ralbidv 3112	. . . . . . . . 9 ⊢ (𝑦 = (𝑢 ∩ 𝑌) → (∀𝑠 ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ 𝑌)) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 ((𝑢 ∩ 𝑠) ∩ 𝑌) ≠ ∅))
70	64, 69	sylan9bb 510	. . . . . . . 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 345	. . . . . . 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 5333	. . . . . 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 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
74	11	adantr 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐹 ∈ (Fil‘𝑋))
75	3	sselda 3921	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
76		fclsopn 23165	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∀𝑡 ∈ 𝐹 (𝑢 ∩ 𝑡) ≠ ∅))))
77	76	baibd 540	. . . . . . 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 835	. . . . . 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 278	. . 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 3903	. . . 4 ⊢ (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fClus 𝐹) ∧ 𝑥 ∈ 𝑌))
83	82	biancomi 463	. . 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 2736	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 ‘cfv 6433 (class class class)co 7275 ↾_t crest 17131 fBascfbas 20585 TopOnctopon 22059 Filcfil 22996 fClus cfcls 23087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-en 8734 df-fin 8737 df-fi 9170 df-rest 17133 df-topgen 17154 df-fbas 20594 df-fg 20595 df-top 22043 df-topon 22060 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-fil 22997 df-fcls 23092

This theorem is referenced by: relcmpcmet 24482

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