Theorem islly2 21287

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 21275	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2	1	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3		simplr 792	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Locally 𝐴)
4	2	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Top)
5		islly2.2	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	topopn 20711	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
7	4, 6	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝑋 ∈ 𝐽)
8		simpr 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
9		llyi 21277	. . . . . 6 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑋 ∈ 𝐽 ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
10	3, 7, 8, 9	syl3anc 1326	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
11		3simpc 1060	. . . . . 6 ⊢ ((𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
12	11	reximi 3011	. . . . 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 2966	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
15	2, 14	jca 554	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
16		simprl 794	. . 3 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17		elssuni 4467	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
18	17, 5	syl6sseqr 3652	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
19	18	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
20		ssralv 3666	. . . . . . 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 799	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23		simplrl 800	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑧 ∈ 𝐽)
24		simprl 794	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝐽)
25		inopn 20704	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ 𝐽 ∧ 𝑢 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ 𝐽)
26	22, 23, 24, 25	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ 𝐽)
27		inss1 3833	. . . . . . . . . . . . 13 ⊢ (𝑧 ∩ 𝑢) ⊆ 𝑧
28		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
29	28	elpw2 4828	. . . . . . . . . . . . 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 3798	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
33		simplrr 801	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑧)
34		simprrl 804	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑢)
35	33, 34	elind 3798	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧 ∩ 𝑢))
36		inss2 3834	. . . . . . . . . . . . 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 20969	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑧 ∩ 𝑢) ⊆ 𝑢 ∧ 𝑢 ∈ 𝐽) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
39	22, 37, 24, 38	syl3anc 1326	. . . . . . . . . . 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		elrestr 16089	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢))
41	22, 24, 23, 40	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢))
42		simprrr 805	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t 𝑢) ∈ 𝐴)
43		restlly.1	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
44	43	ralrimivva 2971	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴)
45	44	ad3antrrr 766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴)
46		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → (𝑗 ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t 𝑥))
47	46	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → ((𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
48	47	raleqbi1dv 3146	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → (∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ∀𝑥 ∈ (𝐽 ↾t 𝑢)((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
49	48	rspcv 3305	. . . . . . . . . . . . 13 ⊢ ((𝐽 ↾t 𝑢) ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴 → ∀𝑥 ∈ (𝐽 ↾t 𝑢)((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
50	42, 45, 49	sylc 65	. . . . . . . . . . . 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)
51		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑢) ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)))
52	51	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → (((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
53	52	rspcv 3305	. . . . . . . . . . . 12 ⊢ ((𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢) → (∀𝑥 ∈ (𝐽 ↾t 𝑢)((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴 → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
54	41, 50, 53	sylc 65	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)
55	39, 54	eqeltrrd 2702	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)
56		eleq2 2690	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ (𝑧 ∩ 𝑢)))
57		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝐽 ↾t 𝑣) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
58	57	eleq1d 2686	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑣) ∈ 𝐴 ↔ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
59	56, 58	anbi12d 747	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴) ↔ (𝑦 ∈ (𝑧 ∩ 𝑢) ∧ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)))
60	59	rspcev 3309	. . . . . . . . . 10 ⊢ (((𝑧 ∩ 𝑢) ∈ (𝐽 ∩ 𝒫 𝑧) ∧ (𝑦 ∈ (𝑧 ∩ 𝑢) ∧ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
61	32, 35, 55, 60	syl12anc 1324	. . . . . . . . 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))
62	61	rexlimdvaa 3032	. . . . . . . 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)))
63	62	anassrs 680	. . . . . . 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)))
64	63	ralimdva 2962	. . . . . 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)))
65	21, 64	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)))
66	65	ralrimdva 2969	. . . 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)))
67	66	impr 649	. . 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))
68		islly 21271	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝐽 ∀𝑦 ∈ 𝑧 ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)))
69	16, 67, 68	sylanbrc 698	. 2 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
70	15, 69	impbida 877	1 ⊢ (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  (class class class)co 6650   ↾_t crest 16081  Topctop 20698  Locally clly 21267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083  df-top 20699  df-lly 21269

This theorem is referenced by: (None)