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Theorem llyidm 23385

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 23369	. . . 4 ⊢ (𝑗 ∈ Locally Locally 𝐴 → 𝑗 ∈ Top)
2		llyi 23371	. . . . . . 7 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾_t 𝑢) ∈ Locally 𝐴))
3		simprr3 1221	. . . . . . . . 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 770	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾_t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ 𝑗)
5		ssidd 4001	. . . . . . . . . 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 1131	. . . . . . . . . . . 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 23074	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑢 ∈ (𝑗 ↾_t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
9	7, 4, 8	syl2anc 583	. . . . . . . . . 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 712	. . . . . . . . 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 1220	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾_t 𝑢) ∈ Locally 𝐴))) → 𝑦 ∈ 𝑢)
12		llyi 23371	. . . . . . . . 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 1369	. . . . . . . 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 23074	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑣 ∈ (𝑗 ↾_t 𝑢) ↔ (𝑣 ∈ 𝑗 ∧ 𝑣 ⊆ 𝑢)))
15	7, 4, 14	syl2anc 583	. . . . . . . . . . 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 252	. . . . . . . . . 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 770	. . . . . . . . . . . . 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 1219	. . . . . . . . . . . . . . 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 1219	. . . . . . . . . . . . . . . 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 3988	. . . . . . . . . . . . . 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 4603	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥)
24	22, 23	sylibr 233	. . . . . . . . . . . . 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 4190	. . . . . . . . . . . 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 1220	. . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 23062	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ Top ∧ 𝑣 ⊆ 𝑢 ∧ 𝑢 ∈ 𝑗) → ((𝑗 ↾_t 𝑢) ↾_t 𝑣) = (𝑗 ↾_t 𝑣))
30	27, 19, 28, 29	syl3anc 1369	. . . . . . . . . . . . 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 1221	. . . . . . . . . . . . 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 2830	. . . . . . . . . . . 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 602	. . . . . . . . 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 3160	. . . . . . . 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 3157	. . . . . 6 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾_t 𝑣) ∈ 𝐴))
39	38	3expb 1118	. . . . 5 ⊢ ((𝑗 ∈ Locally Locally 𝐴 ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾_t 𝑣) ∈ 𝐴))
40	39	ralrimivva 3196	. . . 4 ⊢ (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ (𝑗 ↾_t 𝑣) ∈ 𝐴))
41		islly 23365	. . . 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 582	. . 3 ⊢ (𝑗 ∈ Locally Locally 𝐴 → 𝑗 ∈ Locally 𝐴)
43	42	ssriv 3982	. 2 ⊢ Locally Locally 𝐴 ⊆ Locally 𝐴
44		llyrest 23382	. . . . 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 23369	. . . . . 6 ⊢ (𝑗 ∈ Locally 𝐴 → 𝑗 ∈ Top)
47	46	ssriv 3982	. . . . 5 ⊢ Locally 𝐴 ⊆ Top
48	47	a1i 11	. . . 4 ⊢ (⊤ → Locally 𝐴 ⊆ Top)
49	45, 48	restlly 23380	. . 3 ⊢ (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
50	49	mptru 1541	. 2 ⊢ Locally 𝐴 ⊆ Locally Locally 𝐴
51	43, 50	eqssi 3994	1 ⊢ Locally Locally 𝐴 = Locally 𝐴

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 ∩ cin 3944 ⊆ wss 3945 𝒫 cpw 4598 (class class class)co 7414 ↾_t crest 17395 Topctop 22788 Locally clly 23361

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-en 8958 df-fin 8961 df-fi 9428 df-rest 17397 df-topgen 17418 df-top 22789 df-topon 22806 df-bases 22842 df-lly 23363

This theorem is referenced by: (None)

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