Theorem subislly 21505

Description: The property of a subspace being locally 𝐴. (Contributed by Mario Carneiro, 10-Mar-2015.)

Assertion

Ref	Expression
subislly	⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))

Distinct variable groups:   𝑥,𝑢,𝑦,𝐴   𝑢,𝐵,𝑥,𝑦   𝑢,𝐽,𝑥,𝑦   𝑢,𝑉,𝑥,𝑦

Proof of Theorem subislly

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resttop 21185	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝐽 ↾t 𝐵) ∈ Top)
2		islly 21492	. . . 4 ⊢ ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ((𝐽 ↾t 𝐵) ∈ Top ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
3	2	baib 525	. . 3 ⊢ ((𝐽 ↾t 𝐵) ∈ Top → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑧 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
4	1, 3	syl 17	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑧 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
5		vex 3354	. . . . 5 ⊢ 𝑥 ∈ V
6	5	inex1 4933	. . . 4 ⊢ (𝑥 ∩ 𝐵) ∈ V
7	6	a1i 11	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐵) ∈ V)
8		elrest 16296	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝑧 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐵)))
9		simpr 471	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → 𝑧 = (𝑥 ∩ 𝐵))
10	9	raleqdv 3293	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → (∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
11		elin 3947	. . . . . . . . 9 ⊢ (𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧) ↔ (𝑤 ∈ (𝐽 ↾t 𝐵) ∧ 𝑤 ∈ 𝒫 𝑧))
12	11	anbi1i 610	. . . . . . . 8 ⊢ ((𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧) ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)) ↔ ((𝑤 ∈ (𝐽 ↾t 𝐵) ∧ 𝑤 ∈ 𝒫 𝑧) ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
13		anass 459	. . . . . . . 8 ⊢ (((𝑤 ∈ (𝐽 ↾t 𝐵) ∧ 𝑤 ∈ 𝒫 𝑧) ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)) ↔ (𝑤 ∈ (𝐽 ↾t 𝐵) ∧ (𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴))))
14	12, 13	bitri 264	. . . . . . 7 ⊢ ((𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧) ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)) ↔ (𝑤 ∈ (𝐽 ↾t 𝐵) ∧ (𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴))))
15	14	rexbii2 3187	. . . . . 6 ⊢ (∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∃𝑤 ∈ (𝐽 ↾t 𝐵)(𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
16		vex 3354	. . . . . . . . 9 ⊢ 𝑢 ∈ V
17	16	inex1 4933	. . . . . . . 8 ⊢ (𝑢 ∩ 𝐵) ∈ V
18	17	a1i 11	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐽) → (𝑢 ∩ 𝐵) ∈ V)
19		elrest 16296	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝑤 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝑤 = (𝑢 ∩ 𝐵)))
20	19	ad2antrr 705	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (𝑤 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝑤 = (𝑢 ∩ 𝐵)))
21		3anass 1080	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝒫 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ (𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
22		simpr 471	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑤 = (𝑢 ∩ 𝐵))
23		simpllr 760	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑧 = (𝑥 ∩ 𝐵))
24	22, 23	sseq12d 3783	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ⊆ 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵)))
25		selpw 4304	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝑧 ↔ 𝑤 ⊆ 𝑧)
26		inss2 3982	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝐵) ⊆ 𝐵
27	26	biantru 519	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵))
28		ssin 3983	. . . . . . . . . . 11 ⊢ (((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵) ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
29	27, 28	bitri 264	. . . . . . . . . 10 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
30	24, 25, 29	3bitr4g 303	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ∈ 𝒫 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ 𝑥))
31	22	eleq2d 2836	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (𝑢 ∩ 𝐵)))
32		inss2 3982	. . . . . . . . . . . . 13 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
33		simplr 752	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑦 ∈ (𝑥 ∩ 𝐵))
34	32, 33	sseldi 3750	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑦 ∈ 𝐵)
35	34	biantrud 521	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵)))
36		elin 3947	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑢 ∩ 𝐵) ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵))
37	35, 36	syl6bbr 278	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (𝑢 ∩ 𝐵)))
38	31, 37	bitr4d 271	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑢))
39	22	oveq2d 6809	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝐽 ↾t 𝐵) ↾t 𝑤) = ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)))
40		simp-4l 768	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝐽 ∈ Top)
41	26	a1i 11	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑢 ∩ 𝐵) ⊆ 𝐵)
42		simplr 752	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → 𝐵 ∈ 𝑉)
43	42	ad2antrr 705	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝐵 ∈ 𝑉)
44		restabs 21190	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
45	40, 41, 43, 44	syl3anc 1476	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
46	39, 45	eqtrd 2805	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝐽 ↾t 𝐵) ↾t 𝑤) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
47	46	eleq1d 2835	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴 ↔ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴))
48	30, 38, 47	3anbi123d 1547	. . . . . . . 8 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝑤 ∈ 𝒫 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
49	21, 48	syl5bbr 274	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)) ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
50	18, 20, 49	rexxfr2d 5011	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (∃𝑤 ∈ (𝐽 ↾t 𝐵)(𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)) ↔ ∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
51	15, 50	syl5bb 272	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
52	51	ralbidva 3134	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → (∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
53	10, 52	bitrd 268	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → (∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
54	7, 8, 53	ralxfr2d 5010	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (∀𝑧 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
55	4, 54	bitrd 268	1 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  𝒫 cpw 4297  (class class class)co 6793   ↾_t crest 16289  Topctop 20918  Locally clly 21488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473  df-rest 16291  df-topgen 16312  df-top 20919  df-bases 20971  df-lly 21490

This theorem is referenced by:  iccllysconn  31570