Theorem dislly 23391

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 768	. . . . 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 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	snelpw 5408	. . . . . 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 4630	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
7	6	a1i 11	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {𝑥})
8		llyi 23368	. . . . 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 1373	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴))
10		simpr1 1195	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑦 ⊆ {𝑥})
11		simpr2 1196	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑥 ∈ 𝑦)
12	11	snssd 4776	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
13	10, 12	eqssd 3967	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑦 = {𝑥})
14	13	oveq2d 7406	. . . . . . . 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 774	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑋 ∈ 𝑉)
16		simplr 768	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑥 ∈ 𝑋)
17	16	snssd 4776	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
18		restdis 23072	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑥} ⊆ 𝑋) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
19	15, 17, 18	syl2anc 584	. . . . . . . 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 2765	. . . . . . 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 1197	. . . . . . 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 2830	. . . . . 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 3140	. . . 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 3126	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) → ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴)
27		distop 22889	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
28	27	adantr 480	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Top)
29		elpwi 4573	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
30	29	adantl 481	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝑦 ⊆ 𝑋)
31		ssralv 4018	. . . . . . . 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 770	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ 𝑦)
34	33	snssd 4776	. . . . . . . . . . . . 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 3960	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
37		vsnex 5392	. . . . . . . . . . . . 13 ⊢ {𝑥} ∈ V
38	37	elpw 4570	. . . . . . . . . . . 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 4570	. . . . . . . . . . . 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 4166	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦))
43		snidg 4627	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ {𝑥})
44	43	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ {𝑥})
45		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑋 ∈ 𝑉)
46	45, 36, 18	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
47		simprr 772	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
48	46, 47	eqeltrd 2829	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴)
49		eleq2 2818	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑥} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ {𝑥}))
50		oveq2 7398	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑥} → (𝒫 𝑋 ↾t 𝑢) = (𝒫 𝑋 ↾t {𝑥}))
51	50	eleq1d 2814	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑥} → ((𝒫 𝑋 ↾t 𝑢) ∈ 𝐴 ↔ (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴))
52	49, 51	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑢 = {𝑥} → ((𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴) ↔ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴)))
53	52	rspcev 3591	. . . . . . . . . 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 836	. . . . . . . . 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 3146	. . . . . . 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 652	. . . 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 3126	. . 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 23362	. . 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 583	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Locally 𝐴)
63	26, 62	impbida 800	1 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  {csn 4592  (class class class)co 7390   ↾_t crest 17390  Topctop 22787  Locally clly 23358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-en 8922  df-fin 8925  df-fi 9369  df-rest 17392  df-topgen 17413  df-top 22788  df-topon 22805  df-bases 22840  df-lly 23360

This theorem is referenced by:  disllycmp  23392  dis1stc  23393