Theorem islly2 21510

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 21498	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2	1	adantl 473	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3		simplr 809	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Locally 𝐴)
4	2	adantr 472	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Top)
5		islly2.2	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	topopn 20934	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
7	4, 6	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝑋 ∈ 𝐽)
8		simpr 479	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
9		llyi 21500	. . . . . 6 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑋 ∈ 𝐽 ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
10	3, 7, 8, 9	syl3anc 1477	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
11		3simpc 1147	. . . . . 6 ⊢ ((𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
12	11	reximi 3150	. . . . 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 3105	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
15	2, 14	jca 555	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
16		simprl 811	. . 3 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17		elssuni 4620	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
18	17, 5	syl6sseqr 3794	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
19	18	adantl 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
20		ssralv 3808	. . . . . . 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 817	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23		simplrl 819	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑧 ∈ 𝐽)
24		simprl 811	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝐽)
25		inopn 20927	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ 𝐽 ∧ 𝑢 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ 𝐽)
26	22, 23, 24, 25	syl3anc 1477	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ 𝐽)
27		inss1 3977	. . . . . . . . . . . . 13 ⊢ (𝑧 ∩ 𝑢) ⊆ 𝑧
28		vex 3344	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
29	28	elpw2 4978	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑢) ∈ 𝒫 𝑧 ↔ (𝑧 ∩ 𝑢) ⊆ 𝑧)
30	27, 29	mpbir 221	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝑢) ∈ 𝒫 𝑧
31	30	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ 𝒫 𝑧)
32	26, 31	elind 3942	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
33		simplrr 820	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑧)
34		simprrl 823	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑢)
35	33, 34	elind 3942	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧 ∩ 𝑢))
36		inss2 3978	. . . . . . . . . . . . 13 ⊢ (𝑧 ∩ 𝑢) ⊆ 𝑢
37	36	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ⊆ 𝑢)
38		restabs 21192	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑧 ∩ 𝑢) ⊆ 𝑢 ∧ 𝑢 ∈ 𝐽) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
39	22, 37, 24, 38	syl3anc 1477	. . . . . . . . . . 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)))
40		oveq2 6823	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑢) ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)))
41	40	eleq1d 2825	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → (((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
42		oveq1 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → (𝑗 ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t 𝑥))
43	42	eleq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → ((𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
44	43	raleqbi1dv 3286	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → (∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ∀𝑥 ∈ (𝐽 ↾t 𝑢)((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
45		restlly.1	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
46	45	ralrimivva 3110	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴)
47	46	ad3antrrr 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴)
48		simprrr 824	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t 𝑢) ∈ 𝐴)
49	44, 47, 48	rspcdva 3456	. . . . . . . . . . . 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)
50		elrestr 16312	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢))
51	22, 24, 23, 50	syl3anc 1477	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢))
52	41, 49, 51	rspcdva 3456	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)
53	39, 52	eqeltrrd 2841	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)
54		eleq2 2829	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ (𝑧 ∩ 𝑢)))
55		oveq2 6823	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝐽 ↾t 𝑣) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
56	55	eleq1d 2825	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑣) ∈ 𝐴 ↔ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
57	54, 56	anbi12d 749	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴) ↔ (𝑦 ∈ (𝑧 ∩ 𝑢) ∧ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)))
58	57	rspcev 3450	. . . . . . . . . 10 ⊢ (((𝑧 ∩ 𝑢) ∈ (𝐽 ∩ 𝒫 𝑧) ∧ (𝑦 ∈ (𝑧 ∩ 𝑢) ∧ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
59	32, 35, 53, 58	syl12anc 1475	. . . . . . . . 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))
60	59	rexlimdvaa 3171	. . . . . . . 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)))
61	60	anassrs 683	. . . . . . 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)))
62	61	ralimdva 3101	. . . . . 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)))
63	21, 62	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)))
64	63	ralrimdva 3108	. . . 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)))
65	64	impr 650	. . 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))
66		islly 21494	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝐽 ∀𝑦 ∈ 𝑧 ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)))
67	16, 65, 66	sylanbrc 701	. 2 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
68	15, 67	impbida 913	1 ⊢ (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  ∀wral 3051  ∃wrex 3052   ∩ cin 3715   ⊆ wss 3716  𝒫 cpw 4303  ∪ cuni 4589  (class class class)co 6815   ↾_t crest 16304  Topctop 20921  Locally clly 21490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-rest 16306  df-top 20922  df-lly 21492

This theorem is referenced by: (None)