Theorem restnlly 23781

Description: If the property 𝐴 passes to open subspaces, then a space is n-locally 𝐴 iff it is locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)

Hypothesis

Ref	Expression
restlly.1	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)

Assertion

Ref	Expression
restnlly	⊢ (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)

Distinct variable groups:   𝑥,𝑗,𝐴   𝜑,𝑗,𝑥

Proof of Theorem restnlly

Dummy variables 𝑘 𝑠 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nllytop 23772	. . . . . 6 ⊢ (𝑘 ∈ 𝑛-Locally 𝐴 → 𝑘 ∈ Top)
2	1	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Top)
3		nlly2i 23775	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑛-Locally 𝐴 ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → ∃𝑠 ∈ 𝒫 𝑦∃𝑥 ∈ 𝑘 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))
4	3	3adant1l 1195	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → ∃𝑠 ∈ 𝒫 𝑦∃𝑥 ∈ 𝑘 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))
5		simprl 783	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ∈ 𝑘)
6		simprr2 1241	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ⊆ 𝑠)
7		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑦)
8	7	elpwid 4566	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝑦)
9	6, 8	sstrd 3941	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ⊆ 𝑦)
10		velpw 4562	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 𝑦 ↔ 𝑥 ⊆ 𝑦)
11	9, 10	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ∈ 𝒫 𝑦)
12	5, 11	elind 4146	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘 ∩ 𝒫 𝑦))
13		simprr1 1240	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑢 ∈ 𝑥)
14		simpll1 1231	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴))
15	14, 1	simpl2im 513	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑘 ∈ Top)
16		restabs 23463	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ Top ∧ 𝑥 ⊆ 𝑠 ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑘 ↾t 𝑠) ↾t 𝑥) = (𝑘 ↾t 𝑥))
17	15, 6, 7, 16	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → ((𝑘 ↾t 𝑠) ↾t 𝑥) = (𝑘 ↾t 𝑥))
18		dfss2 3917	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑠 ↔ (𝑥 ∩ 𝑠) = 𝑥)
19	6, 18	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑥 ∩ 𝑠) = 𝑥)
20		elrestr 17579	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ Top ∧ 𝑠 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ 𝑘) → (𝑥 ∩ 𝑠) ∈ (𝑘 ↾t 𝑠))
21	15, 7, 5, 20	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑥 ∩ 𝑠) ∈ (𝑘 ↾t 𝑠))
22	19, 21	eqeltrrd 2862	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘 ↾t 𝑠))
23		eleq2 2850	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑘 ↾t 𝑠) → (𝑥 ∈ 𝑗 ↔ 𝑥 ∈ (𝑘 ↾t 𝑠)))
24		oveq1 7419	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑘 ↾t 𝑠) → (𝑗 ↾t 𝑥) = ((𝑘 ↾t 𝑠) ↾t 𝑥))
25	24	eleq1d 2846	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑘 ↾t 𝑠) → ((𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴))
26	23, 25	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑘 ↾t 𝑠) → ((𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴) ↔ (𝑥 ∈ (𝑘 ↾t 𝑠) → ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴)))
27	14	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝜑)
28		restlly.1	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
29	28	expr 462	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴))
30	29	ralrimiva 3155	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 (𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴))
31	27, 30	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → ∀𝑗 ∈ 𝐴 (𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴))
32		simprr3 1242	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑘 ↾t 𝑠) ∈ 𝐴)
33	26, 31, 32	rspcdva 3578	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘 ↾t 𝑠) → ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴))
34	22, 33	mpd 16	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴)
35	17, 34	eqeltrrd 2862	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑘 ↾t 𝑥) ∈ 𝐴)
36	12, 13, 35	jca32 525	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴)))
37	36	ex 418	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴)) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))))
38	37	reximdv2 3173	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → (∃𝑥 ∈ 𝑘 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴)))
39	38	rexlimdva 3164	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → (∃𝑠 ∈ 𝒫 𝑦∃𝑥 ∈ 𝑘 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴)))
40	4, 39	mpd 16	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))
41	40	3expb 1138	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ (𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦)) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))
42	41	ralrimivva 3206	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → ∀𝑦 ∈ 𝑘 ∀𝑢 ∈ 𝑦 ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))
43		islly 23767	. . . . 5 ⊢ (𝑘 ∈ Locally 𝐴 ↔ (𝑘 ∈ Top ∧ ∀𝑦 ∈ 𝑘 ∀𝑢 ∈ 𝑦 ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴)))
44	2, 42, 43	sylanbrc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Locally 𝐴)
45	44	ex 418	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑛-Locally 𝐴 → 𝑘 ∈ Locally 𝐴))
46	45	ssrdv 3937	. 2 ⊢ (𝜑 → 𝑛-Locally 𝐴 ⊆ Locally 𝐴)
47		llyssnlly 23777	. . 3 ⊢ Locally 𝐴 ⊆ 𝑛-Locally 𝐴
48	47	a1i 11	. 2 ⊢ (𝜑 → Locally 𝐴 ⊆ 𝑛-Locally 𝐴)
49	46, 48	eqssd 3948	1 ⊢ (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  (class class class)co 7412   ↾_t crest 17571  Topctop 23191  Locally clly 23763  𝑛-Locally cnlly 23764

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-rest 17573  df-top 23192  df-nei 23396  df-lly 23765  df-nlly 23766

This theorem is used by:  loclly  23786  hausnlly  23792