Theorem dislly 23462

Description: The discrete space 𝒫 𝑋 is locally 𝐴 if and only if every singleton space has property 𝐴. (Contributed by Mario Carneiro, 20-Mar-2015.)

Assertion

Ref	Expression
dislly	⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝑥,𝑋

Proof of Theorem dislly

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

Step	Hyp	Ref	Expression
1		simplr 769	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝒫 𝑋 ∈ Locally 𝐴)
2		simpr 484	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
3		vex 3434	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	snelpw 5398	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
5	2, 4	sylib 218	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
6		vsnid 4608	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
7	6	a1i 11	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {𝑥})
8		llyi 23439	. . . . 5 ⊢ ((𝒫 𝑋 ∈ Locally 𝐴 ∧ {𝑥} ∈ 𝒫 𝑋 ∧ 𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴))
9	1, 5, 7, 8	syl3anc 1374	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴))
10		simpr1 1196	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑦 ⊆ {𝑥})
11		simpr2 1197	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑥 ∈ 𝑦)
12	11	snssd 4731	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
13	10, 12	eqssd 3940	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑦 = {𝑥})
14	13	oveq2d 7383	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → (𝒫 𝑋 ↾t 𝑦) = (𝒫 𝑋 ↾t {𝑥}))
15		simplll 775	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑋 ∈ 𝑉)
16		simplr 769	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑥 ∈ 𝑋)
17	16	snssd 4731	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
18		restdis 23143	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑥} ⊆ 𝑋) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
19	15, 17, 18	syl2anc 585	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
20	14, 19	eqtrd 2772	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → (𝒫 𝑋 ↾t 𝑦) = 𝒫 {𝑥})
21		simpr3 1198	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)
22	20, 21	eqeltrrd 2838	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
23	22	ex 412	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴) → 𝒫 {𝑥} ∈ 𝐴))
24	23	rexlimdvw 3144	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴) → 𝒫 {𝑥} ∈ 𝐴))
25	9, 24	mpd 15	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝒫 {𝑥} ∈ 𝐴)
26	25	ralrimiva 3130	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) → ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴)
27		distop 22960	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
28	27	adantr 480	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Top)
29		elpwi 4549	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
30	29	adantl 481	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝑦 ⊆ 𝑋)
31		ssralv 3991	. . . . . . . 8 ⊢ (𝑦 ⊆ 𝑋 → (∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴 → ∀𝑥 ∈ 𝑦 𝒫 {𝑥} ∈ 𝐴))
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴 → ∀𝑥 ∈ 𝑦 𝒫 {𝑥} ∈ 𝐴))
33		simprl 771	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ 𝑦)
34	33	snssd 4731	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
35	30	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑦 ⊆ 𝑋)
36	34, 35	sstrd 3933	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
37		vsnex 5378	. . . . . . . . . . . . 13 ⊢ {𝑥} ∈ V
38	37	elpw 4546	. . . . . . . . . . . 12 ⊢ ({𝑥} ∈ 𝒫 𝑋 ↔ {𝑥} ⊆ 𝑋)
39	36, 38	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑋)
40	37	elpw 4546	. . . . . . . . . . . 12 ⊢ ({𝑥} ∈ 𝒫 𝑦 ↔ {𝑥} ⊆ 𝑦)
41	34, 40	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑦)
42	39, 41	elind 4141	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦))
43		snidg 4605	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ {𝑥})
44	43	ad2antrl 729	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ {𝑥})
45		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑋 ∈ 𝑉)
46	45, 36, 18	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
47		simprr 773	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
48	46, 47	eqeltrd 2837	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴)
49		eleq2 2826	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑥} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ {𝑥}))
50		oveq2 7375	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑥} → (𝒫 𝑋 ↾t 𝑢) = (𝒫 𝑋 ↾t {𝑥}))
51	50	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑥} → ((𝒫 𝑋 ↾t 𝑢) ∈ 𝐴 ↔ (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴))
52	49, 51	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑢 = {𝑥} → ((𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴) ↔ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴)))
53	52	rspcev 3565	. . . . . . . . . 10 ⊢ (({𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦) ∧ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴))
54	42, 44, 48, 53	syl12anc 837	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴))
55	54	expr 456	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑦) → (𝒫 {𝑥} ∈ 𝐴 → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴)))
56	55	ralimdva 3150	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝑦 𝒫 {𝑥} ∈ 𝐴 → ∀𝑥 ∈ 𝑦 ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴)))
57	32, 56	syld 47	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴 → ∀𝑥 ∈ 𝑦 ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴)))
58	57	imp 406	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑥 ∈ 𝑦 ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴))
59	58	an32s 653	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) ∧ 𝑦 ∈ 𝒫 𝑋) → ∀𝑥 ∈ 𝑦 ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴))
60	59	ralrimiva 3130	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑦 ∈ 𝒫 𝑋∀𝑥 ∈ 𝑦 ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴))
61		islly 23433	. . 3 ⊢ (𝒫 𝑋 ∈ Locally 𝐴 ↔ (𝒫 𝑋 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑋∀𝑥 ∈ 𝑦 ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴)))
62	28, 60, 61	sylanbrc 584	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Locally 𝐴)
63	26, 62	impbida 801	1 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  (class class class)co 7367   ↾_t crest 17383  Topctop 22858  Locally clly 23429

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-en 8894  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-lly 23431

This theorem is referenced by:  disllycmp  23463  dis1stc  23464