Theorem llyidm 23645

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 23629	. . . 4 ⊢ (𝑗 ∈ Locally Locally 𝐴 → 𝑗 ∈ Top)
2		llyi 23631	. . . . . . 7 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))
3		simprr3 1242	. . . . . . . . 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 782	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ 𝑗)
5		ssidd 3960	. . . . . . . . . 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 1151	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → 𝑗 ∈ Top)
7	6	adantr 485	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑗 ∈ Top)
8		restopn2 23334	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
9	7, 4, 8	syl2anc 595	. . . . . . . . . 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 725	. . . . . . . . 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 1241	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑦 ∈ 𝑢)
12		llyi 23631	. . . . . . . . 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 1398	. . . . . . . 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 23334	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑣 ∈ (𝑗 ↾t 𝑢) ↔ (𝑣 ∈ 𝑗 ∧ 𝑣 ⊆ 𝑢)))
15	7, 4, 14	syl2anc 595	. . . . . . . . . . 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 487	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑗 ∧ 𝑣 ⊆ 𝑢) → 𝑣 ∈ 𝑗)
17	15, 16	biimtrdi 256	. . . . . . . . . 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 782	. . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . 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 4567	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥)
24	22, 23	sylibr 237	. . . . . . . . . . . . 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 4153	. . . . . . . . . . . 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 1241	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . 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 23322	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ Top ∧ 𝑣 ⊆ 𝑢 ∧ 𝑢 ∈ 𝑗) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
30	27, 19, 28, 29	syl3anc 1398	. . . . . . . . . . . . 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 1242	. . . . . . . . . . . . 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 2864	. . . . . . . . . . . 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 524	. . . . . . . . . . 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 417	. . . . . . . . . 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 614	. . . . . . . . 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 3175	. . . . . . . 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 16	. . . . . . 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 3172	. . . . . 6 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
39	38	3expb 1138	. . . . 5 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
40	39	ralrimivva 3208	. . . 4 ⊢ (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
41		islly 23625	. . . 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 594	. . 3 ⊢ (𝑗 ∈ Locally Locally 𝐴 → 𝑗 ∈ Locally 𝐴)
43	42	ssriv 3941	. 2 ⊢ Locally Locally 𝐴 ⊆ Locally 𝐴
44		llyrest 23642	. . . . 5 ⊢ ((𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
45	44	adantl 486	. . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
46		llytop 23629	. . . . . 6 ⊢ (𝑗 ∈ Locally 𝐴 → 𝑗 ∈ Top)
47	46	ssriv 3941	. . . . 5 ⊢ Locally 𝐴 ⊆ Top
48	47	a1i 11	. . . 4 ⊢ (⊤ → Locally 𝐴 ⊆ Top)
49	45, 48	restlly 23640	. . 3 ⊢ (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
50	49	mptru 1577	. 2 ⊢ Locally 𝐴 ⊆ Locally Locally 𝐴
51	43, 50	eqssi 3953	1 ⊢ Locally Locally 𝐴 = Locally 𝐴

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562  (class class class)co 7410   ↾_t crest 17468  Topctop 23050  Locally clly 23621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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 7859  df-1st 7982  df-2nd 7983  df-en 8940  df-fin 8943  df-fi 9367  df-rest 17470  df-topgen 17491  df-top 23051  df-topon 23068  df-bases 23103  df-lly 23623

This theorem is referenced by: (None)