Theorem dis2ndc 22070

Description: A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)

Assertion

Ref	Expression
dis2ndc	⊢ (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)

Proof of Theorem dis2ndc

Dummy variables 𝑤 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ctex 8526	. 2 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V)
2		pwexr 7489	. 2 ⊢ (𝒫 𝑋 ∈ 2ndω → 𝑋 ∈ V)
3		snex 5334	. . . . . . . 8 ⊢ {𝑥} ∈ V
4	3	2a1i 12	. . . . . . 7 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 → {𝑥} ∈ V))
5		vex 3499	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6	5	sneqr 4773	. . . . . . . . 9 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
7		sneq 4579	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
8	6, 7	impbii 211	. . . . . . . 8 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
9	8	2a1i 12	. . . . . . 7 ⊢ (𝑋 ∈ V → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
10	4, 9	dom2lem 8551	. . . . . 6 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1→V)
11		f1f1orn 6628	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1→V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1-onto→ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
12	10, 11	syl 17	. . . . 5 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1-onto→ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
13		f1oeng 8530	. . . . 5 ⊢ ((𝑋 ∈ V ∧ (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1-onto→ran (𝑥 ∈ 𝑋 ↦ {𝑥})) → 𝑋 ≈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
14	12, 13	mpdan 685	. . . 4 ⊢ (𝑋 ∈ V → 𝑋 ≈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
15		domen1 8661	. . . 4 ⊢ (𝑋 ≈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) → (𝑋 ≼ ω ↔ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
16	14, 15	syl 17	. . 3 ⊢ (𝑋 ∈ V → (𝑋 ≼ ω ↔ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
17		distop 21605	. . . . . . 7 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
18		simpr 487	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
19	5	snelpw 5340	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
20	18, 19	sylib 220	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
21	20	fmpttd 6881	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋⟶𝒫 𝑋)
22	21	frnd 6523	. . . . . . 7 ⊢ (𝑋 ∈ V → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋)
23		elpwi 4550	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
24	23	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑦 ⊆ 𝑋)
25		simprr 771	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ 𝑦)
26	24, 25	sseldd 3970	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ 𝑋)
27		eqidd 2824	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} = {𝑧})
28		sneq 4579	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
29	28	rspceeqv 3640	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ {𝑧} = {𝑧}) → ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
30	26, 27, 29	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
31		snex 5334	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
32		eqid 2823	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑥})
33	32	elrnmpt 5830	. . . . . . . . . . 11 ⊢ ({𝑧} ∈ V → ({𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥}))
34	31, 33	ax-mp 5	. . . . . . . . . 10 ⊢ ({𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
35	30, 34	sylibr 236	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
36		vsnid 4604	. . . . . . . . . 10 ⊢ 𝑧 ∈ {𝑧}
37	36	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ {𝑧})
38	25	snssd 4744	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} ⊆ 𝑦)
39		eleq2 2903	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑧} → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ {𝑧}))
40		sseq1 3994	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑧} → (𝑤 ⊆ 𝑦 ↔ {𝑧} ⊆ 𝑦))
41	39, 40	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑤 = {𝑧} → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦) ↔ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)))
42	41	rspcev 3625	. . . . . . . . 9 ⊢ (({𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∧ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
43	35, 37, 38, 42	syl12anc 834	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
44	43	ralrimivva 3193	. . . . . . 7 ⊢ (𝑋 ∈ V → ∀𝑦 ∈ 𝒫 𝑋∀𝑧 ∈ 𝑦 ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
45		basgen2 21599	. . . . . . 7 ⊢ ((𝒫 𝑋 ∈ Top ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋∀𝑧 ∈ 𝑦 ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦)) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
46	17, 22, 44, 45	syl3anc 1367	. . . . . 6 ⊢ (𝑋 ∈ V → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
47	46	adantr 483	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
48	46, 17	eqeltrd 2915	. . . . . . 7 ⊢ (𝑋 ∈ V → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ Top)
49		tgclb 21580	. . . . . . 7 ⊢ (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ Top)
50	48, 49	sylibr 236	. . . . . 6 ⊢ (𝑋 ∈ V → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases)
51		2ndci 22058	. . . . . 6 ⊢ ((ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ 2ndω)
52	50, 51	sylan 582	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ 2ndω)
53	47, 52	eqeltrrd 2916	. . . 4 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → 𝒫 𝑋 ∈ 2ndω)
54		is2ndc 22056	. . . . . 6 ⊢ (𝒫 𝑋 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋))
55		vex 3499	. . . . . . . . 9 ⊢ 𝑏 ∈ V
56		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
57	56, 19	sylib 220	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
58		simplrr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → (topGen‘𝑏) = 𝒫 𝑋)
59	57, 58	eleqtrrd 2918	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (topGen‘𝑏))
60		vsnid 4604	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ {𝑥}
61		tg2 21575	. . . . . . . . . . . . 13 ⊢ (({𝑥} ∈ (topGen‘𝑏) ∧ 𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))
62	59, 60, 61	sylancl 588	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))
63		simprrl 779	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → 𝑥 ∈ 𝑦)
64	63	snssd 4744	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} ⊆ 𝑦)
65		simprrr 780	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → 𝑦 ⊆ {𝑥})
66	64, 65	eqssd 3986	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} = 𝑦)
67		simprl 769	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → 𝑦 ∈ 𝑏)
68	66, 67	eqeltrd 2915	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} ∈ 𝑏)
69	62, 68	rexlimddv 3293	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝑏)
70	69	fmpttd 6881	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋⟶𝑏)
71	70	frnd 6523	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝑏)
72		ssdomg 8557	. . . . . . . . 9 ⊢ (𝑏 ∈ V → (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝑏 → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏))
73	55, 71, 72	mpsyl 68	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏)
74		simprl 769	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → 𝑏 ≼ ω)
75		domtr 8564	. . . . . . . 8 ⊢ ((ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
76	73, 74, 75	syl2anc 586	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
77	76	rexlimdva2 3289	. . . . . 6 ⊢ (𝑋 ∈ V → (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
78	54, 77	syl5bi 244	. . . . 5 ⊢ (𝑋 ∈ V → (𝒫 𝑋 ∈ 2ndω → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
79	78	imp 409	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝒫 𝑋 ∈ 2ndω) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
80	53, 79	impbida 799	. . 3 ⊢ (𝑋 ∈ V → (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
81	16, 80	bitrd 281	. 2 ⊢ (𝑋 ∈ V → (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
82	1, 2, 81	pm5.21nii 382	1 ⊢ (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  𝒫 cpw 4541  {csn 4569   class class class wbr 5068   ↦ cmpt 5148  ran crn 5558  –1-1→wf1 6354  –1-1-onto→wf1o 6356  ‘cfv 6357  ωcom 7582   ≈ cen 8508   ≼ cdom 8509  topGenctg 16713  Topctop 21503  TopBasesctb 21555  2^ndωc2ndc 22048

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 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-er 8291  df-en 8512  df-dom 8513  df-topgen 16719  df-top 21504  df-bases 21556  df-2ndc 22050

This theorem is referenced by: (None)