Theorem dis2ndc 23585

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 8959	. 2 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V)
2		pwexr 7763	. 2 ⊢ (𝒫 𝑋 ∈ 2ndω → 𝑋 ∈ V)
3		vsnex 5407	. . . . . . . 8 ⊢ {𝑥} ∈ V
4	3	2a1i 12	. . . . . . 7 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 → {𝑥} ∈ V))
5		vex 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6	5	sneqr 4809	. . . . . . . . 9 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
7		sneq 4604	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
8	6, 7	impbii 212	. . . . . . . 8 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
9	8	2a1i 12	. . . . . . 7 ⊢ (𝑋 ∈ V → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
10	4, 9	dom2lem 8988	. . . . . 6 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1→V)
11		f1f1orn 6833	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1→V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1-onto→ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
12	10, 11	syl 18	. . . . 5 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1-onto→ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
13		f1oeng 8966	. . . . 5 ⊢ ((𝑋 ∈ V ∧ (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋–1-1-onto→ran (𝑥 ∈ 𝑋 ↦ {𝑥})) → 𝑋 ≈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
14	12, 13	mpdan 699	. . . 4 ⊢ (𝑋 ∈ V → 𝑋 ≈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
15		domen1 9106	. . . 4 ⊢ (𝑋 ≈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) → (𝑋 ≼ ω ↔ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
16	14, 15	syl 18	. . 3 ⊢ (𝑋 ∈ V → (𝑋 ≼ ω ↔ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
17		distop 23120	. . . . . . 7 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
18	5	snelpw 5427	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
19	18	bilani 509	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
20	19	fmpttd 7111	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋⟶𝒫 𝑋)
21	20	frnd 6715	. . . . . . 7 ⊢ (𝑋 ∈ V → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋)
22		elpwi 4574	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
23	22	ad2antrl 740	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑦 ⊆ 𝑋)
24		simprr 784	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ 𝑦)
25	23, 24	sseldd 3946	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ 𝑋)
26		eqidd 2770	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} = {𝑧})
27		sneq 4604	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
28	27	rspceeqv 3613	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ {𝑧} = {𝑧}) → ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
29	25, 26, 28	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
30		vsnex 5407	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
31		eqid 2769	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑥})
32	31	elrnmpt 5949	. . . . . . . . . . 11 ⊢ ({𝑧} ∈ V → ({𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥}))
33	30, 32	ax-mp 5	. . . . . . . . . 10 ⊢ ({𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
34	29, 33	sylibr 237	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
35		vsnid 4634	. . . . . . . . . 10 ⊢ 𝑧 ∈ {𝑧}
36	35	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ {𝑧})
37	24	snssd 4757	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} ⊆ 𝑦)
38		eleq2 2858	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑧} → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ {𝑧}))
39		sseq1 3970	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑧} → (𝑤 ⊆ 𝑦 ↔ {𝑧} ⊆ 𝑦))
40	38, 39	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑤 = {𝑧} → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦) ↔ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)))
41	40	rspcev 3590	. . . . . . . . 9 ⊢ (({𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∧ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
42	34, 36, 37, 41	syl12anc 849	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
43	42	ralrimivva 3214	. . . . . . 7 ⊢ (𝑋 ∈ V → ∀𝑦 ∈ 𝒫 𝑋∀𝑧 ∈ 𝑦 ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
44		basgen2 23114	. . . . . . 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)
45	17, 21, 43, 44	syl3anc 1396	. . . . . 6 ⊢ (𝑋 ∈ V → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
46	45	adantr 485	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
47	45, 17	eqeltrd 2869	. . . . . . 7 ⊢ (𝑋 ∈ V → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ Top)
48		tgclb 23095	. . . . . . 7 ⊢ (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ Top)
49	47, 48	sylibr 237	. . . . . 6 ⊢ (𝑋 ∈ V → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases)
50		2ndci 23573	. . . . . 6 ⊢ ((ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ 2ndω)
51	49, 50	sylan 591	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ 2ndω)
52	46, 51	eqeltrrd 2870	. . . 4 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → 𝒫 𝑋 ∈ 2ndω)
53		is2ndc 23571	. . . . . 6 ⊢ (𝒫 𝑋 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋))
54		vex 3467	. . . . . . . . 9 ⊢ 𝑏 ∈ V
55	18	bilani 509	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
56		simplrr 789	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → (topGen‘𝑏) = 𝒫 𝑋)
57	55, 56	eleqtrrd 2872	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (topGen‘𝑏))
58		vsnid 4634	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ {𝑥}
59		tg2 23090	. . . . . . . . . . . . 13 ⊢ (({𝑥} ∈ (topGen‘𝑏) ∧ 𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))
60	57, 58, 59	sylancl 597	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))
61		simprrl 792	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → 𝑥 ∈ 𝑦)
62	61	snssd 4757	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} ⊆ 𝑦)
63		simprrr 793	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → 𝑦 ⊆ {𝑥})
64	62, 63	eqssd 3962	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} = 𝑦)
65		simprl 782	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → 𝑦 ∈ 𝑏)
66	64, 65	eqeltrd 2869	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} ∈ 𝑏)
67	60, 66	rexlimddv 3178	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝑏)
68	67	fmpttd 7111	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋⟶𝑏)
69	68	frnd 6715	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝑏)
70		ssdomg 8996	. . . . . . . . 9 ⊢ (𝑏 ∈ V → (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝑏 → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏))
71	54, 69, 70	mpsyl 69	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏)
72		simprl 782	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → 𝑏 ≼ ω)
73		domtr 9003	. . . . . . . 8 ⊢ ((ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
74	71, 72, 73	syl2anc 595	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
75	74	rexlimdva2 3174	. . . . . 6 ⊢ (𝑋 ∈ V → (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
76	53, 75	biimtrid 245	. . . . 5 ⊢ (𝑋 ∈ V → (𝒫 𝑋 ∈ 2ndω → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
77	76	imp 411	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝒫 𝑋 ∈ 2ndω) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
78	52, 77	impbida 812	. . 3 ⊢ (𝑋 ∈ V → (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
79	16, 78	bitrd 282	. 2 ⊢ (𝑋 ∈ V → (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
80	1, 2, 79	pm5.21nii 381	1 ⊢ (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567  {csn 4594   class class class wbr 5113   ↦ cmpt 5196  ran crn 5663  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  ωcom 7861   ≈ cen 8939   ≼ cdom 8940  topGenctg 17489  Topctop 23018  TopBasesctb 23070  2^ndωc2ndc 23563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8693  df-en 8943  df-dom 8944  df-topgen 17495  df-top 23019  df-bases 23071  df-2ndc 23565

This theorem is referenced by: (None)