Theorem dislly 22105

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 767	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝒫 𝑋 ∈ Locally 𝐴)
2		simpr 487	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
3		vex 3497	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	snelpw 5338	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
5	2, 4	sylib 220	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
6		vsnid 4602	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
7	6	a1i 11	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {𝑥})
8		llyi 22082	. . . . 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 1367	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴))
10		simpr1 1190	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑦 ⊆ {𝑥})
11		simpr2 1191	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑥 ∈ 𝑦)
12	11	snssd 4742	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
13	10, 12	eqssd 3984	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑦 = {𝑥})
14	13	oveq2d 7172	. . . . . . . 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 773	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑋 ∈ 𝑉)
16		simplr 767	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑥 ∈ 𝑋)
17	16	snssd 4742	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
18		restdis 21786	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑥} ⊆ 𝑋) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
19	15, 17, 18	syl2anc 586	. . . . . . . 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 2856	. . . . . . 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 1192	. . . . . . 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 2914	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
23	22	ex 415	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴) → 𝒫 {𝑥} ∈ 𝐴))
24	23	rexlimdvw 3290	. . . 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 3182	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) → ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴)
27		distop 21603	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
28	27	adantr 483	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Top)
29		elpwi 4548	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
30	29	adantl 484	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝑦 ⊆ 𝑋)
31		ssralv 4033	. . . . . . . 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 769	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ 𝑦)
34	33	snssd 4742	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
35	30	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑦 ⊆ 𝑋)
36	34, 35	sstrd 3977	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
37		snex 5332	. . . . . . . . . . . . 13 ⊢ {𝑥} ∈ V
38	37	elpw 4543	. . . . . . . . . . . 12 ⊢ ({𝑥} ∈ 𝒫 𝑋 ↔ {𝑥} ⊆ 𝑋)
39	36, 38	sylibr 236	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑋)
40	37	elpw 4543	. . . . . . . . . . . 12 ⊢ ({𝑥} ∈ 𝒫 𝑦 ↔ {𝑥} ⊆ 𝑦)
41	34, 40	sylibr 236	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑦)
42	39, 41	elind 4171	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦))
43		snidg 4599	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ {𝑥})
44	43	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ {𝑥})
45		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑋 ∈ 𝑉)
46	45, 36, 18	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
47		simprr 771	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
48	46, 47	eqeltrd 2913	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴)
49		eleq2 2901	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑥} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ {𝑥}))
50		oveq2 7164	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑥} → (𝒫 𝑋 ↾t 𝑢) = (𝒫 𝑋 ↾t {𝑥}))
51	50	eleq1d 2897	. . . . . . . . . . . 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 3623	. . . . . . . . . 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 834	. . . . . . . . 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 459	. . . . . . . 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 3177	. . . . . . 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 409	. . . . 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 650	. . . 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 3182	. . 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 22076	. . 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 585	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Locally 𝐴)
63	26, 62	impbida 799	1 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  {csn 4567  (class class class)co 7156   ↾_t crest 16694  Topctop 21501  Locally clly 22072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-oadd 8106  df-er 8289  df-en 8510  df-fin 8513  df-fi 8875  df-rest 16696  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-lly 22074

This theorem is referenced by:  disllycmp  22106  dis1stc  22107