Theorem dis2ndc 23443

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 7708	. 2 ⊢ (𝒫 𝑋 ∈ 2ndω → 𝑋 ∈ V)
3		vsnex 5364	. . . . . . . 8 ⊢ {𝑥} ∈ V
4	3	2a1i 12	. . . . . . 7 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 → {𝑥} ∈ V))
5		vex 3435	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6	5	sneqr 4771	. . . . . . . . 9 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
7		sneq 4565	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
8	6, 7	impbii 210	. . . . . . . 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 6778	. . . . . 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 693	. . . 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 22978	. . . . . . 7 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
18	5	snelpw 5384	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
19	18	bilani 505	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
20	19	fmpttd 7056	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋⟶𝒫 𝑋)
21	20	frnd 6663	. . . . . . 7 ⊢ (𝑋 ∈ V → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋)
22		elpwi 4536	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
23	22	ad2antrl 734	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑦 ⊆ 𝑋)
24		simprr 778	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ 𝑦)
25	23, 24	sseldd 3916	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ 𝑋)
26		eqidd 2740	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} = {𝑧})
27		sneq 4565	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
28	27	rspceeqv 3583	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ {𝑧} = {𝑧}) → ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
29	25, 26, 28	syl2anc 590	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → ∃𝑥 ∈ 𝑋 {𝑧} = {𝑥})
30		vsnex 5364	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
31		eqid 2739	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑥})
32	31	elrnmpt 5900	. . . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥}))
35		vsnid 4595	. . . . . . . . . 10 ⊢ 𝑧 ∈ {𝑧}
36	35	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → 𝑧 ∈ {𝑧})
37	24	snssd 4718	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → {𝑧} ⊆ 𝑦)
38		eleq2 2828	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑧} → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ {𝑧}))
39		sseq1 3940	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑧} → (𝑤 ⊆ 𝑦 ↔ {𝑧} ⊆ 𝑦))
40	38, 39	anbi12d 638	. . . . . . . . . 10 ⊢ (𝑤 = {𝑧} → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦) ↔ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)))
41	40	rspcev 3560	. . . . . . . . 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 842	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
43	42	ralrimivva 3182	. . . . . . 7 ⊢ (𝑋 ∈ V → ∀𝑦 ∈ 𝒫 𝑋∀𝑧 ∈ 𝑦 ∃𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑥})(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
44		basgen2 22972	. . . . . . 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 1379	. . . . . 6 ⊢ (𝑋 ∈ V → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
46	45	adantr 481	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) = 𝒫 𝑋)
47	45, 17	eqeltrd 2839	. . . . . . 7 ⊢ (𝑋 ∈ V → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ Top)
48		tgclb 22953	. . . . . . 7 ⊢ (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ Top)
49	47, 48	sylibr 235	. . . . . 6 ⊢ (𝑋 ∈ V → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases)
50		2ndci 23431	. . . . . 6 ⊢ ((ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ∈ TopBases ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ 2ndω)
51	49, 50	sylan 586	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑋 ↦ {𝑥})) ∈ 2ndω)
52	46, 51	eqeltrrd 2840	. . . 4 ⊢ ((𝑋 ∈ V ∧ ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω) → 𝒫 𝑋 ∈ 2ndω)
53		is2ndc 23429	. . . . . 6 ⊢ (𝒫 𝑋 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋))
54		vex 3435	. . . . . . . . 9 ⊢ 𝑏 ∈ V
55	18	bilani 505	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
56		simplrr 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → (topGen‘𝑏) = 𝒫 𝑋)
57	55, 56	eleqtrrd 2842	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (topGen‘𝑏))
58		vsnid 4595	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ {𝑥}
59		tg2 22948	. . . . . . . . . . . . 13 ⊢ (({𝑥} ∈ (topGen‘𝑏) ∧ 𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))
60	57, 58, 59	sylancl 592	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))
61		simprrl 786	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → 𝑥 ∈ 𝑦)
62	61	snssd 4718	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} ⊆ 𝑦)
63		simprrr 787	. . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ {𝑥}))) → {𝑥} = 𝑦)
65		simprl 776	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 3146	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝑏)
68	67	fmpttd 7056	. . . . . . . . . 10 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → (𝑥 ∈ 𝑋 ↦ {𝑥}):𝑋⟶𝑏)
69	68	frnd 6663	. . . . . . . . 9 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝑏)
70		ssdomg 8937	. . . . . . . . 9 ⊢ (𝑏 ∈ V → (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ⊆ 𝑏 → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏))
71	54, 69, 70	mpsyl 68	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏)
72		simprl 776	. . . . . . . 8 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → 𝑏 ≼ ω)
73		domtr 8944	. . . . . . . 8 ⊢ ((ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
74	71, 72, 73	syl2anc 590	. . . . . . 7 ⊢ (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
75	74	rexlimdva2 3142	. . . . . 6 ⊢ (𝑋 ∈ V → (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
76	53, 75	biimtrid 243	. . . . 5 ⊢ (𝑋 ∈ V → (𝒫 𝑋 ∈ 2ndω → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω))
77	76	imp 407	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝒫 𝑋 ∈ 2ndω) → ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω)
78	52, 77	impbida 806	. . 3 ⊢ (𝑋 ∈ V → (ran (𝑥 ∈ 𝑋 ↦ {𝑥}) ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
79	16, 78	bitrd 280	. 2 ⊢ (𝑋 ∈ V → (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
80	1, 2, 79	pm5.21nii 379	1 ⊢ (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  𝒫 cpw 4529  {csn 4555   class class class wbr 5072   ↦ cmpt 5153  ran crn 5619  –1-1→wf1 6482  –1-1-onto→wf1o 6484  ‘cfv 6485  ωcom 7806   ≈ cen 8880   ≼ cdom 8881  topGenctg 17391  Topctop 22876  TopBasesctb 22928  2^ndωc2ndc 23421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-er 8633  df-en 8884  df-dom 8885  df-topgen 17397  df-top 22877  df-bases 22929  df-2ndc 23423

This theorem is referenced by: (None)