Theorem llyidm 23466

Description: Idempotence of the "locally" predicate, i.e. being "locally 𝐴 " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llyidm	⊢ Locally Locally 𝐴 = Locally 𝐴

Proof of Theorem llyidm

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

Step	Hyp	Ref	Expression
1		llytop 23450	. . . 4 ⊢ (𝑗 ∈ Locally Locally 𝐴 → 𝑗 ∈ Top)
2		llyi 23452	. . . . . . 7 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))
3		simprr3 1225	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → (𝑗 ↾t 𝑢) ∈ Locally 𝐴)
4		simprl 771	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ 𝑗)
5		ssidd 3946	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑢 ⊆ 𝑢)
6	1	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → 𝑗 ∈ Top)
7	6	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 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 Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
10	4, 5, 9	mpbir2and 714	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ (𝑗 ↾t 𝑢))
11		simprr2 1224	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑦 ∈ 𝑢)
12		llyi 23452	. . . . . . . . 9 ⊢ (((𝑗 ↾t 𝑢) ∈ Locally 𝐴 ∧ 𝑢 ∈ (𝑗 ↾t 𝑢) ∧ 𝑦 ∈ 𝑢) → ∃𝑣 ∈ (𝑗 ↾t 𝑢)(𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))
13	3, 10, 11, 12	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗 ↾t 𝑢)(𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))
14		restopn2 23155	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑣 ∈ (𝑗 ↾t 𝑢) ↔ (𝑣 ∈ 𝑗 ∧ 𝑣 ⊆ 𝑢)))
15	7, 4, 14	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗 ↾t 𝑢) ↔ (𝑣 ∈ 𝑗 ∧ 𝑣 ⊆ 𝑢)))
16		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑗 ∧ 𝑣 ⊆ 𝑢) → 𝑣 ∈ 𝑗)
17	15, 16	biimtrdi 253	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗 ↾t 𝑢) → 𝑣 ∈ 𝑗))
18		simprl 771	. . . . . . . . . . . . 13 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝑗)
19		simprr1 1223	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝑢)
20		simprr1 1223	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑢 ⊆ 𝑥)
21	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 ⊆ 𝑥)
22	19, 21	sstrd 3933	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝑥)
23		velpw 4547	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥)
24	22, 23	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
25	18, 24	elind 4141	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝑗 ∩ 𝒫 𝑥))
26		simprr2 1224	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦 ∈ 𝑣)
27	7	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
28		simplrl 777	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 ∈ 𝑗)
29		restabs 23143	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ Top ∧ 𝑣 ⊆ 𝑢 ∧ 𝑢 ∈ 𝑗) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
30	27, 19, 28, 29	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
31		simprr3 1225	. . . . . . . . . . . . 13 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴)
32	30, 31	eqeltrrd 2838	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗 ↾t 𝑣) ∈ 𝐴)
33	25, 26, 32	jca32 515	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))
34	33	ex 412	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → ((𝑣 ∈ 𝑗 ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))))
35	17, 34	syland 604	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → ((𝑣 ∈ (𝑗 ↾t 𝑢) ∧ (𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))))
36	35	reximdv2 3148	. . . . . . . 8 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → (∃𝑣 ∈ (𝑗 ↾t 𝑢)(𝑣 ⊆ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))
37	13, 36	mpd 15	. . . . . . 7 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
38	2, 37	rexlimddv 3145	. . . . . 6 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
39	38	3expb 1121	. . . . 5 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
40	39	ralrimivva 3181	. . . 4 ⊢ (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
41		islly 23446	. . . 4 ⊢ (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))
42	1, 40, 41	sylanbrc 584	. . 3 ⊢ (𝑗 ∈ Locally Locally 𝐴 → 𝑗 ∈ Locally 𝐴)
43	42	ssriv 3926	. 2 ⊢ Locally Locally 𝐴 ⊆ Locally 𝐴
44		llyrest 23463	. . . . 5 ⊢ ((𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
45	44	adantl 481	. . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
46		llytop 23450	. . . . . 6 ⊢ (𝑗 ∈ Locally 𝐴 → 𝑗 ∈ Top)
47	46	ssriv 3926	. . . . 5 ⊢ Locally 𝐴 ⊆ Top
48	47	a1i 11	. . . 4 ⊢ (⊤ → Locally 𝐴 ⊆ Top)
49	45, 48	restlly 23461	. . 3 ⊢ (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
50	49	mptru 1549	. 2 ⊢ Locally 𝐴 ⊆ Locally Locally 𝐴
51	43, 50	eqssi 3939	1 ⊢ Locally Locally 𝐴 = Locally 𝐴

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  (class class class)co 7361   ↾_t crest 17377  Topctop 22871  Locally clly 23442

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-lly 23444

This theorem is referenced by: (None)