Theorem dis2ndc 23404

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 8900	. 2 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V)
2		pwexr 7710	. 2 ⊢ (𝒫 𝑋 ∈ 2ndω → 𝑋 ∈ V)
3		vsnex 5379	. . . . . . . 8 ⊢ {𝑥} ∈ V
4	3	2a1i 12	. . . . . . 7 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 → {𝑥} ∈ V))
5		vex 3444	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6	5	sneqr 4796	. . . . . . . . 9 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
7		sneq 4590	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
8	6, 7	impbii 209	. . . . . . . 8 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
9	8	2a1i 12	. . . . . . 7 ⊢ (𝑋 ∈ V → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
10	4, 9	dom2lem 8929	. . . . . 6 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1→V)
11		f1f1orn 6785	. . . . . 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 8907	. . . . 5 ⊢ ((𝑋 ∈ V ∧ (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1-onto→ran (𝑥 ∈ 𝑋 ↦ {𝑥})) → 𝑋 ≈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
14	12, 13	mpdan 687	. . . 4 ⊢ (𝑋 ∈ V → 𝑋 ≈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
15		domen1 9047	. . . 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 22939	. . . . . . 7 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
18		simpr 484	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
19	5	snelpw 5393	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
20	18, 19	sylib 218	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
21	20	fmpttd 7060	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋⟶𝒫 𝑋)
22	21	frnd 6670	. . . . . . 7 ⊢ (𝑋 ∈ V → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋)
23		elpwi 4561	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
24	23	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑦 ⊆ 𝑋)
25		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ 𝑦)
26	24, 25	sseldd 3934	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ 𝑋)
27		eqidd 2737	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} = {𝑧})
28		sneq 4590	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
29	28	rspceeqv 3599	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ {𝑧} = {𝑧}) → ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
30	26, 27, 29	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
31		vsnex 5379	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
32		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑥})
33	32	elrnmpt 5907	. . . . . . . . . . 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 234	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
36		vsnid 4620	. . . . . . . . . 10 ⊢ 𝑧 ∈ {𝑧}
37	36	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ {𝑧})
38	25	snssd 4765	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} ⊆ 𝑦)
39		eleq2 2825	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑧} → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ {𝑧}))
40		sseq1 3959	. . . . . . . . . . 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 3576	. . . . . . . . 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 836	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
44	43	ralrimivva 3179	. . . . . . 7 ⊢ (𝑋 ∈ V → ∀𝑦 ∈ 𝒫 𝑋∀𝑧 ∈ 𝑦 ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
45		basgen2 22933	. . . . . . 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 1373	. . . . . 6 ⊢ (𝑋 ∈ V → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
47	46	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
48	46, 17	eqeltrd 2836	. . . . . . 7 ⊢ (𝑋 ∈ V → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ Top)
49		tgclb 22914	. . . . . . 7 ⊢ (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ Top)
50	48, 49	sylibr 234	. . . . . 6 ⊢ (𝑋 ∈ V → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases)
51		2ndci 23392	. . . . . 6 ⊢ ((ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ 2ndω)
52	50, 51	sylan 580	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ 2ndω)
53	47, 52	eqeltrrd 2837	. . . 4 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → 𝒫 𝑋 ∈ 2ndω)
54		is2ndc 23390	. . . . . 6 ⊢ (𝒫 𝑋 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋))
55		vex 3444	. . . . . . . . 9 ⊢ 𝑏 ∈ V
56		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
57	56, 19	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
58		simplrr 777	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → (topGen‘𝑏) = 𝒫 𝑋)
59	57, 58	eleqtrrd 2839	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (topGen‘𝑏))
60		vsnid 4620	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ {𝑥}
61		tg2 22909	. . . . . . . . . . . . 13 ⊢ (({𝑥} ∈ (topGen‘𝑏) ∧ 𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))
62	59, 60, 61	sylancl 586	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))
63		simprrl 780	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → 𝑥 ∈ 𝑦)
64	63	snssd 4765	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} ⊆ 𝑦)
65		simprrr 781	. . . . . . . . . . . . . 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 3951	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} = 𝑦)
67		simprl 770	. . . . . . . . . . . . 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 2836	. . . . . . . . . . . 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 3143	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝑏)
70	69	fmpttd 7060	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋⟶𝑏)
71	70	frnd 6670	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝑏)
72		ssdomg 8937	. . . . . . . . 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 770	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → 𝑏 ≼ ω)
75		domtr 8944	. . . . . . . 8 ⊢ ((ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
76	73, 74, 75	syl2anc 584	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
77	76	rexlimdva2 3139	. . . . . 6 ⊢ (𝑋 ∈ V → (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
78	54, 77	biimtrid 242	. . . . 5 ⊢ (𝑋 ∈ V → (𝒫 𝑋 ∈ 2ndω → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
79	78	imp 406	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝒫 𝑋 ∈ 2ndω) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
80	53, 79	impbida 800	. . 3 ⊢ (𝑋 ∈ V → (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
81	16, 80	bitrd 279	. 2 ⊢ (𝑋 ∈ V → (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
82	1, 2, 81	pm5.21nii 378	1 ⊢ (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554  {csn 4580   class class class wbr 5098   ↦ cmpt 5179  ran crn 5625  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492  ωcom 7808   ≈ cen 8880   ≼ cdom 8881  topGenctg 17357  Topctop 22837  TopBasesctb 22889  2^ndωc2ndc 23382

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-er 8635  df-en 8884  df-dom 8885  df-topgen 17363  df-top 22838  df-bases 22890  df-2ndc 23384

This theorem is referenced by: (None)