Theorem llyidm 23715

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 23699	. . . 4 ⊢ (𝑗 ∈ Locally Locally 𝐴 → 𝑗 ∈ Top)
2		llyi 23701	. . . . . . 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 783	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ 𝑗)
5		ssidd 3954	. . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ Locally 𝐴))) → 𝑗 ∈ Top)
8		restopn2 23403	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
9	7, 4, 8	syl2anc 596	. . . . . . . . . 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 726	. . . . . . . . 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 23701	. . . . . . . . 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 23403	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑣 ∈ (𝑗 ↾t 𝑢) ↔ (𝑣 ∈ 𝑗 ∧ 𝑣 ⊆ 𝑢)))
15	7, 4, 14	syl2anc 596	. . . . . . . . . . 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 488	. . . . . . . . . . 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 783	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . 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 4562	. . . . . . . . . . . . . 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 4146	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . 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 23391	. . . . . . . . . . . . . 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 2861	. . . . . . . . . . . 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 525	. . . . . . . . . . 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 418	. . . . . . . . . 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 615	. . . . . . . . 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 3172	. . . . . . . 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 3169	. . . . . 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 3205	. . . 4 ⊢ (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
41		islly 23695	. . . 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 595	. . 3 ⊢ (𝑗 ∈ Locally Locally 𝐴 → 𝑗 ∈ Locally 𝐴)
43	42	ssriv 3935	. 2 ⊢ Locally Locally 𝐴 ⊆ Locally 𝐴
44		llyrest 23712	. . . . 5 ⊢ ((𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
45	44	adantl 487	. . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Locally 𝐴)
46		llytop 23699	. . . . . 6 ⊢ (𝑗 ∈ Locally 𝐴 → 𝑗 ∈ Top)
47	46	ssriv 3935	. . . . 5 ⊢ Locally 𝐴 ⊆ Top
48	47	a1i 11	. . . 4 ⊢ (⊤ → Locally 𝐴 ⊆ Top)
49	45, 48	restlly 23710	. . 3 ⊢ (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
50	49	mptru 1577	. 2 ⊢ Locally 𝐴 ⊆ Locally Locally 𝐴
51	43, 50	eqssi 3947	1 ⊢ Locally Locally 𝐴 = Locally 𝐴

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  (class class class)co 7414   ↾_t crest 17506  Topctop 23119  Locally clly 23691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-en 8954  df-fin 8957  df-fi 9382  df-rest 17508  df-topgen 17529  df-top 23120  df-topon 23137  df-bases 23172  df-lly 23693

This theorem is used by: (None)