Theorem dislly 22556

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 765	. . . . 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 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	snelpw 5355	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
5	2, 4	sylib 217	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
6		vsnid 4595	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
7	6	a1i 11	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {𝑥})
8		llyi 22533	. . . . 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 1369	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴))
10		simpr1 1192	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑦 ⊆ {𝑥})
11		simpr2 1193	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑥 ∈ 𝑦)
12	11	snssd 4739	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
13	10, 12	eqssd 3934	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑦 = {𝑥})
14	13	oveq2d 7271	. . . . . . . 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 771	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑋 ∈ 𝑉)
16		simplr 765	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → 𝑥 ∈ 𝑋)
17	16	snssd 4739	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥 ∈ 𝑦 ∧ (𝒫 𝑋 ↾t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
18		restdis 22237	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑥} ⊆ 𝑋) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
19	15, 17, 18	syl2anc 583	. . . . . . . 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 2778	. . . . . . 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 1194	. . . . . . 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 2840	. . . . . 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 3218	. . . 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 3107	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) → ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴)
27		distop 22053	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
28	27	adantr 480	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Top)
29		elpwi 4539	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
30	29	adantl 481	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝑦 ⊆ 𝑋)
31		ssralv 3983	. . . . . . . 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 767	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ 𝑦)
34	33	snssd 4739	. . . . . . . . . . . . 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 3927	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
37		snex 5349	. . . . . . . . . . . . 13 ⊢ {𝑥} ∈ V
38	37	elpw 4534	. . . . . . . . . . . 12 ⊢ ({𝑥} ∈ 𝒫 𝑋 ↔ {𝑥} ⊆ 𝑋)
39	36, 38	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑋)
40	37	elpw 4534	. . . . . . . . . . . 12 ⊢ ({𝑥} ∈ 𝒫 𝑦 ↔ {𝑥} ⊆ 𝑦)
41	34, 40	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑦)
42	39, 41	elind 4124	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦))
43		snidg 4592	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ {𝑥})
44	43	ad2antrl 724	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ {𝑥})
45		simpll 763	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑋 ∈ 𝑉)
46	45, 36, 18	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) = 𝒫 {𝑥})
47		simprr 769	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
48	46, 47	eqeltrd 2839	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴)
49		eleq2 2827	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑥} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ {𝑥}))
50		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑥} → (𝒫 𝑋 ↾t 𝑢) = (𝒫 𝑋 ↾t {𝑥}))
51	50	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑥} → ((𝒫 𝑋 ↾t 𝑢) ∈ 𝐴 ↔ (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴))
52	49, 51	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑢 = {𝑥} → ((𝑥 ∈ 𝑢 ∧ (𝒫 𝑋 ↾t 𝑢) ∈ 𝐴) ↔ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋 ↾t {𝑥}) ∈ 𝐴)))
53	52	rspcev 3552	. . . . . . . . . 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 833	. . . . . . . . 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 3102	. . . . . . 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 648	. . . 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 3107	. . 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 22527	. . 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 582	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Locally 𝐴)
63	26, 62	impbida 797	1 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  (class class class)co 7255   ↾_t crest 17048  Topctop 21950  Locally clly 22523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-en 8692  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-lly 22525

This theorem is referenced by:  disllycmp  22557  dis1stc  22558