Theorem restnlly 23457

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 23448	. . . . . 6 ⊢ (𝑘 ∈ 𝑛-Locally 𝐴 → 𝑘 ∈ Top)
2	1	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Top)
3		nlly2i 23451	. . . . . . . . 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 1178	. . . . . . . 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 771	. . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 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 4551	. . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . 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 4547	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 𝑦 ↔ 𝑥 ⊆ 𝑦)
11	9, 10	sylibr 234	. . . . . . . . . . . . 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 4141	. . . . . . . . . . . 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 1223	. . . . . . . . . . . 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 1214	. . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . 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 23140	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ Top ∧ 𝑥 ⊆ 𝑠 ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑘 ↾t 𝑠) ↾t 𝑥) = (𝑘 ↾t 𝑥))
17	15, 6, 7, 16	syl3anc 1374	. . . . . . . . . . . . 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 3908	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑠 ↔ (𝑥 ∩ 𝑠) = 𝑥)
19	6, 18	sylib 218	. . . . . . . . . . . . . . 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 17382	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ Top ∧ 𝑠 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ 𝑘) → (𝑥 ∩ 𝑠) ∈ (𝑘 ↾t 𝑠))
21	15, 7, 5, 20	syl3anc 1374	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑘 ↾t 𝑠) → (𝑥 ∈ 𝑗 ↔ 𝑥 ∈ (𝑘 ↾t 𝑠)))
24		oveq1 7367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑘 ↾t 𝑠) → (𝑗 ↾t 𝑥) = ((𝑘 ↾t 𝑠) ↾t 𝑥))
25	24	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑘 ↾t 𝑠) → ((𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴))
26	23, 25	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑘 ↾t 𝑠) → ((𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴) ↔ (𝑥 ∈ (𝑘 ↾t 𝑠) → ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴)))
27	14	simpld 494	. . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴))
30	29	ralrimiva 3130	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 (𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴))
31	27, 30	syl 17	. . . . . . . . . . . . . . 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 1225	. . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . . 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 15	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 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 515	. . . . . . . . . . 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 412	. . . . . . . . . 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 3148	. . . . . . . . 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 3139	. . . . . . . 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 15	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))
41	40	3expb 1121	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ (𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦)) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))
42	41	ralrimivva 3181	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → ∀𝑦 ∈ 𝑘 ∀𝑢 ∈ 𝑦 ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))
43		islly 23443	. . . . 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 584	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Locally 𝐴)
45	44	ex 412	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑛-Locally 𝐴 → 𝑘 ∈ Locally 𝐴))
46	45	ssrdv 3928	. 2 ⊢ (𝜑 → 𝑛-Locally 𝐴 ⊆ Locally 𝐴)
47		llyssnlly 23453	. . 3 ⊢ Locally 𝐴 ⊆ 𝑛-Locally 𝐴
48	47	a1i 11	. 2 ⊢ (𝜑 → Locally 𝐴 ⊆ 𝑛-Locally 𝐴)
49	46, 48	eqssd 3940	1 ⊢ (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  (class class class)co 7360   ↾_t crest 17374  Topctop 22868  Locally clly 23439  𝑛-Locally cnlly 23440

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-rest 17376  df-top 22869  df-nei 23073  df-lly 23441  df-nlly 23442

This theorem is referenced by:  loclly  23462  hausnlly  23468