Theorem islly2 22988

Description: An alternative expression for 𝐽 ∈ Locally 𝐴 when 𝐴 passes to open subspaces: A space is locally 𝐴 if every point is contained in an open neighborhood with property 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)

Hypotheses

Ref	Expression
restlly.1	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
islly2.2	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
islly2	⊢ (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))

Distinct variable groups:   𝑢,𝑗,𝑥,𝑦,𝐴   𝑗,𝐽,𝑢,𝑥,𝑦   𝜑,𝑗,𝑢,𝑥,𝑦   𝑢,𝑋,𝑦

Allowed substitution hints:   𝑋(𝑥,𝑗)

Proof of Theorem islly2

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

Step	Hyp	Ref	Expression
1		llytop 22976	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2	1	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Locally 𝐴)
4	2	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Top)
5		islly2.2	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	topopn 22408	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
7	4, 6	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝑋 ∈ 𝐽)
8		simpr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
9		llyi 22978	. . . . . 6 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑋 ∈ 𝐽 ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
10	3, 7, 8, 9	syl3anc 1372	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
11		3simpc 1151	. . . . . 6 ⊢ ((𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
12	11	reximi 3085	. . . . 5 ⊢ (∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
13	10, 12	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
14	13	ralrimiva 3147	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
15	2, 14	jca 513	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
16		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17		elssuni 4942	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
18	17, 5	sseqtrrdi 4034	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
19	18	adantl 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
20		ssralv 4051	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑋 → (∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → ∀𝑦 ∈ 𝑧 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
21	19, 20	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) → (∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → ∀𝑦 ∈ 𝑧 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
22		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23		simplrl 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑧 ∈ 𝐽)
24		simprl 770	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝐽)
25		inopn 22401	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ 𝐽 ∧ 𝑢 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ 𝐽)
26	22, 23, 24, 25	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ 𝐽)
27		vex 3479	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
28		inss1 4229	. . . . . . . . . . . . 13 ⊢ (𝑧 ∩ 𝑢) ⊆ 𝑧
29	27, 28	elpwi2 5347	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝑢) ∈ 𝒫 𝑧
30	29	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ 𝒫 𝑧)
31	26, 30	elind 4195	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
32		simplrr 777	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑧)
33		simprrl 780	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑢)
34	32, 33	elind 4195	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧 ∩ 𝑢))
35		inss2 4230	. . . . . . . . . . . . 13 ⊢ (𝑧 ∩ 𝑢) ⊆ 𝑢
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ⊆ 𝑢)
37		restabs 22669	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑧 ∩ 𝑢) ⊆ 𝑢 ∧ 𝑢 ∈ 𝐽) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
38	22, 36, 24, 37	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
39		oveq2 7417	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑢) ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)))
40	39	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → (((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
41		oveq1 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → (𝑗 ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t 𝑥))
42	41	eleq1d 2819	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → ((𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
43	42	raleqbi1dv 3334	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → (∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ∀𝑥 ∈ (𝐽 ↾t 𝑢)((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
44		restlly.1	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
45	44	ralrimivva 3201	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴)
46	45	ad3antrrr 729	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴)
47		simprrr 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t 𝑢) ∈ 𝐴)
48	43, 46, 47	rspcdva 3614	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ∀𝑥 ∈ (𝐽 ↾t 𝑢)((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴)
49		elrestr 17374	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢))
50	22, 24, 23, 49	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢))
51	40, 48, 50	rspcdva 3614	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)
52	38, 51	eqeltrrd 2835	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)
53		eleq2 2823	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ (𝑧 ∩ 𝑢)))
54		oveq2 7417	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝐽 ↾t 𝑣) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
55	54	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑣) ∈ 𝐴 ↔ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
56	53, 55	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴) ↔ (𝑦 ∈ (𝑧 ∩ 𝑢) ∧ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)))
57	56	rspcev 3613	. . . . . . . . . 10 ⊢ (((𝑧 ∩ 𝑢) ∈ (𝐽 ∩ 𝒫 𝑧) ∧ (𝑦 ∈ (𝑧 ∩ 𝑢) ∧ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
58	31, 34, 52, 57	syl12anc 836	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
59	58	rexlimdvaa 3157	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) → (∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)))
60	59	anassrs 469	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) ∧ 𝑦 ∈ 𝑧) → (∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)))
61	60	ralimdva 3168	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) → (∀𝑦 ∈ 𝑧 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → ∀𝑦 ∈ 𝑧 ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)))
62	21, 61	syld 47	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) → (∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → ∀𝑦 ∈ 𝑧 ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)))
63	62	ralrimdva 3155	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ Top) → (∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → ∀𝑧 ∈ 𝐽 ∀𝑦 ∈ 𝑧 ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)))
64	63	impr 456	. . 3 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ∀𝑧 ∈ 𝐽 ∀𝑦 ∈ 𝑧 ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
65		islly 22972	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝐽 ∀𝑦 ∈ 𝑧 ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)))
66	16, 64, 65	sylanbrc 584	. 2 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
67	15, 66	impbida 800	1 ⊢ (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909  (class class class)co 7409   ↾_t crest 17366  Topctop 22395  Locally clly 22968

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-rest 17368  df-top 22396  df-lly 22970

This theorem is referenced by: (None)