Theorem islly2 22635

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 22623	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2	1	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3		simplr 766	. . . . . 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 22055	. . . . . . 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 22625	. . . . . 6 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑋 ∈ 𝐽 ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
10	3, 7, 8, 9	syl3anc 1370	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ Locally 𝐴) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
11		3simpc 1149	. . . . . 6 ⊢ ((𝑢 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) → (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
12	11	reximi 3178	. . . . 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 3103	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
15	2, 14	jca 512	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
16		simprl 768	. . 3 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17		elssuni 4871	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
18	17, 5	sseqtrrdi 3972	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ 𝑋)
19	18	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ Top) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
20		ssralv 3987	. . . . . . 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 773	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23		simplrl 774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑧 ∈ 𝐽)
24		simprl 768	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝐽)
25		inopn 22048	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ 𝐽 ∧ 𝑢 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ 𝐽)
26	22, 23, 24, 25	syl3anc 1370	. . . . . . . . . . 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 4162	. . . . . . . . . . . . 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 4128	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝑧 ∩ 𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
32		simplrr 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑧)
33		simprrl 778	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑢)
34	32, 33	elind 4128	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧 ∩ 𝑢))
35		inss2 4163	. . . . . . . . . . . . 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 22316	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑧 ∩ 𝑢) ⊆ 𝑢 ∧ 𝑢 ∈ 𝐽) → ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
38	22, 36, 24, 37	syl3anc 1370	. . . . . . . . . . 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 7283	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑢) ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)))
40	39	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑧 ∩ 𝑢) → (((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
41		oveq1 7282	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → (𝑗 ↾t 𝑥) = ((𝐽 ↾t 𝑢) ↾t 𝑥))
42	41	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 ↾t 𝑢) → ((𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽 ↾t 𝑢) ↾t 𝑥) ∈ 𝐴))
43	42	raleqbi1dv 3340	. . . . . . . . . . . . 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 3123	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑥 ∈ 𝑗 (𝑗 ↾t 𝑥) ∈ 𝐴)
46	45	ad3antrrr 727	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . 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 3562	. . . . . . . . . . . 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 17139	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑧 ∩ 𝑢) ∈ (𝐽 ↾t 𝑢))
50	22, 24, 23, 49	syl3anc 1370	. . . . . . . . . . . 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 3562	. . . . . . . . . . 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 2840	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐽 ∈ Top) ∧ (𝑧 ∈ 𝐽 ∧ 𝑦 ∈ 𝑧)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)
53		eleq2 2827	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ (𝑧 ∩ 𝑢)))
54		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝐽 ↾t 𝑣) = (𝐽 ↾t (𝑧 ∩ 𝑢)))
55	54	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝐽 ↾t 𝑣) ∈ 𝐴 ↔ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴))
56	53, 55	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴) ↔ (𝑦 ∈ (𝑧 ∩ 𝑢) ∧ (𝐽 ↾t (𝑧 ∩ 𝑢)) ∈ 𝐴)))
57	56	rspcev 3561	. . . . . . . . . 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 834	. . . . . . . . 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 3214	. . . . . . . 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 3108	. . . . . 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 3106	. . . 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 22619	. . 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 583	. 2 ⊢ ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
67	15, 66	impbida 798	1 ⊢ (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  (class class class)co 7275   ↾_t crest 17131  Topctop 22042  Locally clly 22615

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-rest 17133  df-top 22043  df-lly 22617

This theorem is referenced by: (None)