Theorem nllyidm 23467

Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 23465 to show 𝑛-Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴.) (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllyidm	⊢ Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴

Proof of Theorem nllyidm

Dummy variables 𝑗 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		llytop 23450	. . . 4 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → 𝑗 ∈ Top)
2		llyi 23452	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))
3		simprr3 1225	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴)
4		simprl 771	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ∈ 𝑗)
5		ssidd 3946	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ⊆ 𝑢)
6		simpl1 1193	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Locally 𝑛-Locally 𝐴)
7	6, 1	syl 17	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Top)
8		restopn2 23155	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
9	7, 4, 8	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
10	4, 5, 9	mpbir2and 714	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ∈ (𝑗 ↾t 𝑢))
11		simprr2 1224	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑦 ∈ 𝑢)
12		nlly2i 23454	. . . . . . . . 9 ⊢ (((𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴 ∧ 𝑢 ∈ (𝑗 ↾t 𝑢) ∧ 𝑦 ∈ 𝑢) → ∃𝑣 ∈ 𝒫 𝑢∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))
13	3, 10, 11, 12	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ 𝒫 𝑢∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))
14		restopn2 23155	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑧 ∈ (𝑗 ↾t 𝑢) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢)))
15	7, 4, 14	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑧 ∈ (𝑗 ↾t 𝑢) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢)))
16	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗 ↾t 𝑢) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢)))
17	7	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
18		simpr2l 1234	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧 ∈ 𝑗)
19		simpr31 1265	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦 ∈ 𝑧)
20		opnneip 23097	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ Top ∧ 𝑧 ∈ 𝑗 ∧ 𝑦 ∈ 𝑧) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
21	17, 18, 19, 20	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
22		simpr32 1266	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧 ⊆ 𝑣)
23		simpr1 1196	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑢)
24	23	elpwid 4551	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝑢)
25	4	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 ∈ 𝑗)
26		elssuni 4882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝑗 → 𝑢 ⊆ ∪ 𝑗)
27	25, 26	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 ⊆ ∪ 𝑗)
28	24, 27	sstrd 3933	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ ∪ 𝑗)
29		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑗 = ∪ 𝑗
30	29	ssnei2 23094	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 ∈ Top ∧ 𝑧 ∈ ((nei‘𝑗)‘{𝑦})) ∧ (𝑧 ⊆ 𝑣 ∧ 𝑣 ⊆ ∪ 𝑗)) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
31	17, 21, 22, 28, 30	syl22anc 839	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
32		simprr1 1223	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ⊆ 𝑥)
33	32	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 ⊆ 𝑥)
34	24, 33	sstrd 3933	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝑥)
35		velpw 4547	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥)
36	34, 35	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
37	31, 36	elind 4141	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥))
38		restabs 23143	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ Top ∧ 𝑣 ⊆ 𝑢 ∧ 𝑢 ∈ 𝑗) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
39	17, 24, 25, 38	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
40		simpr33 1267	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴)
41	39, 40	eqeltrrd 2838	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗 ↾t 𝑣) ∈ 𝐴)
42	37, 41	jca 511	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
43	42	3exp2 1356	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑣 ∈ 𝒫 𝑢 → ((𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))))
44	43	imp 406	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → ((𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))))
45	16, 44	sylbid 240	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗 ↾t 𝑢) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))))
46	45	rexlimdv 3137	. . . . . . . . . 10 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))
47	46	expimpd 453	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → ((𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))
48	47	reximdv2 3148	. . . . . . . 8 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (∃𝑣 ∈ 𝒫 𝑢∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴))
49	13, 48	mpd 15	. . . . . . 7 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
50	2, 49	rexlimddv 3145	. . . . . 6 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
51	50	3expb 1121	. . . . 5 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
52	51	ralrimivva 3181	. . . 4 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
53		isnlly 23447	. . . 4 ⊢ (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴))
54	1, 52, 53	sylanbrc 584	. . 3 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → 𝑗 ∈ 𝑛-Locally 𝐴)
55	54	ssriv 3926	. 2 ⊢ Locally 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐴
56		nllyrest 23464	. . . . 5 ⊢ ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
57	56	adantl 481	. . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
58		nllytop 23451	. . . . . 6 ⊢ (𝑗 ∈ 𝑛-Locally 𝐴 → 𝑗 ∈ Top)
59	58	ssriv 3926	. . . . 5 ⊢ 𝑛-Locally 𝐴 ⊆ Top
60	59	a1i 11	. . . 4 ⊢ (⊤ → 𝑛-Locally 𝐴 ⊆ Top)
61	57, 60	restlly 23461	. . 3 ⊢ (⊤ → 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴)
62	61	mptru 1549	. 2 ⊢ 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴
63	55, 62	eqssi 3939	1 ⊢ Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851  ‘cfv 6493  (class class class)co 7361   ↾_t crest 17377  Topctop 22871  neicnei 23075  Locally clly 23442  𝑛-Locally cnlly 23443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  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-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-en 8888  df-fin 8891  df-fi 9318  df-rest 17379  df-topgen 17400  df-top 22872  df-topon 22889  df-bases 22924  df-nei 23076  df-lly 23444  df-nlly 23445

This theorem is referenced by: (None)