Theorem islly2 23474

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 23462	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2	1	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3		simplr 774	. . . . . 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 22896	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
7	4, 6	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝑋 ∈ 𝐽)
8		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
9		llyi 23464	. . . . . 6 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑋 ∈ 𝐽 ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
10	3, 7, 8, 9	syl3anc 1379	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
11		3simpc 1156	. . . . . 6 ⊢ ((𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
12	11	reximi 3078	. . . . 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 3132	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
15	2, 14	jca 516	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
16		simprl 776	. . 3 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17		elssuni 4876	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
18	17, 5	sseqtrrdi 3963	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
19	18	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
20		ssralv 3990	. . . . . . 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 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23		simplrl 782	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑧 ∈ 𝐽)
24		simprl 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝐽)
25		inopn 22889	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ 𝐽 ∧ 𝑢 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ 𝐽)
26	22, 23, 24, 25	syl3anc 1379	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ 𝐽)
27		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
28		inss1 4172	. . . . . . . . . . . . 13 ⊢ (𝑧 ∩ 𝑢) ⊆ 𝑧
29	27, 28	elpwi2 5270	. . . . . . . . . . . 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 4136	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
32		simplrr 783	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑧)
33		simprrl 786	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑢)
34	32, 33	elind 4136	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧 ∩ 𝑢))
35		inss2 4173	. . . . . . . . . . . . 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 23155	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑧 ∩ 𝑢) ⊆ 𝑢 ∧ 𝑢 ∈ 𝐽) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
38	22, 36, 24, 37	syl3anc 1379	. . . . . . . . . . 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 7371	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑢) ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)))
40	39	eleq1d 2825	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → (((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
41		oveq1 7370	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → (𝑗 ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t 𝑥))
42	41	eleq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → ((𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
43	42	raleqbi1dv 3308	. . . . . . . . . . . . 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 3183	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴)
46	45	ad3antrrr 736	. . . . . . . . . . . . 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 787	. . . . . . . . . . . . 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 3568	. . . . . . . . . . . 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 17389	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢))
50	22, 24, 23, 49	syl3anc 1379	. . . . . . . . . . . 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 3568	. . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)
53		eleq2 2829	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ (𝑧 ∩ 𝑢)))
54		oveq2 7371	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝐽 ↾t 𝑣) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
55	54	eleq1d 2825	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑣) ∈ 𝐴 ↔ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
56	53, 55	anbi12d 638	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴) ↔ (𝑦 ∈ (𝑧 ∩ 𝑢) ∧ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)))
57	56	rspcev 3567	. . . . . . . . . 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 842	. . . . . . . . 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 3142	. . . . . . . 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 468	. . . . . . 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 3152	. . . . . 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 3140	. . . 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 455	. . 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 23458	. . 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 589	. 2 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
67	15, 66	impbida 806	1 ⊢ (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4536  ∪ cuni 4845  (class class class)co 7363   ↾_t crest 17381  Topctop 22883  Locally clly 23454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-rest 17383  df-top 22884  df-lly 23456

This theorem is referenced by: (None)