Theorem nllyidm 23432

Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 23430 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 23415	. . . 4 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → 𝑗 ∈ Top)
2		llyi 23417	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))
3		simprr3 1224	. . . . . . . . 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 770	. . . . . . . . . 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 3987	. . . . . . . . . 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 1192	. . . . . . . . . . . 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 23120	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
9	7, 4, 8	syl2anc 584	. . . . . . . . . 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 713	. . . . . . . . 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 1223	. . . . . . . . 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 23419	. . . . . . . . 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 1373	. . . . . . . 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 23120	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑧 ∈ (𝑗 ↾t 𝑢) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢)))
15	7, 4, 14	syl2anc 584	. . . . . . . . . . . . 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 1233	. . . . . . . . . . . . . . . . . 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 1264	. . . . . . . . . . . . . . . . . 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 23062	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ Top ∧ 𝑧 ∈ 𝑗 ∧ 𝑦 ∈ 𝑧) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
21	17, 18, 19, 20	syl3anc 1373	. . . . . . . . . . . . . . . . 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 1265	. . . . . . . . . . . . . . . . 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 1195	. . . . . . . . . . . . . . . . . . 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 4589	. . . . . . . . . . . . . . . . . 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 4918	. . . . . . . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑗 = ∪ 𝑗
30	29	ssnei2 23059	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 ∈ Top ∧ 𝑧 ∈ ((nei‘𝑗)‘{𝑦})) ∧ (𝑧 ⊆ 𝑣 ∧ 𝑣 ⊆ ∪ 𝑗)) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
31	17, 21, 22, 28, 30	syl22anc 838	. . . . . . . . . . . . . . . 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 1222	. . . . . . . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . . . . . 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 4585	. . . . . . . . . . . . . . . . 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 4180	. . . . . . . . . . . . . . 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 23108	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ Top ∧ 𝑣 ⊆ 𝑢 ∧ 𝑢 ∈ 𝑗) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
39	17, 24, 25, 38	syl3anc 1373	. . . . . . . . . . . . . . . 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 1266	. . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . 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 1355	. . . . . . . . . . . . 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 3140	. . . . . . . . . 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 3151	. . . . . . . 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 3148	. . . . . 6 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
51	50	3expb 1120	. . . . 5 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
52	51	ralrimivva 3188	. . . 4 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
53		isnlly 23412	. . . 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 583	. . 3 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → 𝑗 ∈ 𝑛-Locally 𝐴)
55	54	ssriv 3967	. 2 ⊢ Locally 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐴
56		nllyrest 23429	. . . . 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 23416	. . . . . 6 ⊢ (𝑗 ∈ 𝑛-Locally 𝐴 → 𝑗 ∈ Top)
59	58	ssriv 3967	. . . . 5 ⊢ 𝑛-Locally 𝐴 ⊆ Top
60	59	a1i 11	. . . 4 ⊢ (⊤ → 𝑛-Locally 𝐴 ⊆ Top)
61	57, 60	restlly 23426	. . 3 ⊢ (⊤ → 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴)
62	61	mptru 1547	. 2 ⊢ 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴
63	55, 62	eqssi 3980	1 ⊢ Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580  {csn 4606  ∪ cuni 4888  ‘cfv 6536  (class class class)co 7410   ↾_t crest 17439  Topctop 22836  neicnei 23040  Locally clly 23407  𝑛-Locally cnlly 23408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-en 8965  df-fin 8968  df-fi 9428  df-rest 17441  df-topgen 17462  df-top 22837  df-topon 22854  df-bases 22889  df-nei 23041  df-lly 23409  df-nlly 23410

This theorem is referenced by: (None)