Theorem dissnlocfin 23848

Description: The set of singletons is locally finite in the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.)

Hypothesis

Ref	Expression
dissnref.c	⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}

Assertion

Ref	Expression
dissnlocfin	⊢ (𝑋 ∈ 𝑉 → 𝐶 ∈ (LocFin‘𝒫 𝑋))

Distinct variable groups:   𝑢,𝐶,𝑥   𝑢,𝑉,𝑥   𝑢,𝑋,𝑥

Proof of Theorem dissnlocfin

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

Step	Hyp	Ref	Expression
1		distop 23313	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
2		eqidd 2761	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 = 𝑋)
3		snelpwi 5411	. . . . 5 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ 𝒫 𝑋)
4	3	adantl 487	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑧} ∈ 𝒫 𝑋)
5		vsnid 4623	. . . . 5 ⊢ 𝑧 ∈ {𝑧}
6	5	a1i 11	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ {𝑧})
7		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑢(𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋)
8		nfrab1 3431	. . . . . 6 ⊢ Ⅎ𝑢{𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅}
9		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑢{{𝑧}}
10		dissnref.c	. . . . . . . . . 10 ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}
11	10	eqabri 2902	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
12	11	anbi1i 636	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
13		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
14		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → 𝑢 = {𝑥})
15	14	ineq1d 4116	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) = ({𝑥} ∩ {𝑧}))
16		disjsn2 4672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ≠ 𝑧 → ({𝑥} ∩ {𝑧}) = ∅)
17	16	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → ({𝑥} ∩ {𝑧}) = ∅)
18	15, 17	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) = ∅)
19		simp-4r 796	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) ≠ ∅)
20	19	neneqd 2960	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → ¬ (𝑢 ∩ {𝑧}) = ∅)
21	18, 20	pm2.65da 829	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → ¬ 𝑥 ≠ 𝑧)
22		nne 2959	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ≠ 𝑧 ↔ 𝑥 = 𝑧)
23	21, 22	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 = 𝑧)
24	23	sneqd 4595	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} = {𝑧})
25	13, 24	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑧})
26	25	r19.29an 3166	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}) → 𝑢 = {𝑧})
27	26	an32s 665	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) → 𝑢 = {𝑧})
28	27	anasss 472	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅)) → 𝑢 = {𝑧})
29		sneq 4593	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
30	29	rspceeqv 3598	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑢 = {𝑧}) → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
31	30	adantll 727	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
32		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → 𝑢 = {𝑧})
33	32	ineq1d 4116	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) = ({𝑧} ∩ {𝑧}))
34		inidm 4123	. . . . . . . . . . . 12 ⊢ ({𝑧} ∩ {𝑧}) = {𝑧}
35	33, 34	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) = {𝑧})
36		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
37	36	snnz 4736	. . . . . . . . . . . 12 ⊢ {𝑧} ≠ ∅
38	37	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → {𝑧} ≠ ∅)
39	35, 38	eqnetrd 3022	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) ≠ ∅)
40	31, 39	jca 521	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
41	28, 40	impbida 813	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ((∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ 𝑢 = {𝑧}))
42	12, 41	bitrid 286	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ((𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ 𝑢 = {𝑧}))
43		rabid 3432	. . . . . . 7 ⊢ (𝑢 ∈ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ↔ (𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
44		velsn 4599	. . . . . . 7 ⊢ (𝑢 ∈ {{𝑧}} ↔ 𝑢 = {𝑧})
45	42, 43, 44	3bitr4g 317	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → (𝑢 ∈ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ↔ 𝑢 ∈ {{𝑧}}))
46	7, 8, 9, 45	eqrd 3927	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} = {{𝑧}})
47		snfi 9071	. . . . 5 ⊢ {{𝑧}} ∈ Fin
48	46, 47	eqeltrdi 2868	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)
49		eleq2 2849	. . . . . 6 ⊢ (𝑦 = {𝑧} → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ {𝑧}))
50		ineq2 4111	. . . . . . . . 9 ⊢ (𝑦 = {𝑧} → (𝑢 ∩ 𝑦) = (𝑢 ∩ {𝑧}))
51	50	neeq1d 3014	. . . . . . . 8 ⊢ (𝑦 = {𝑧} → ((𝑢 ∩ 𝑦) ≠ ∅ ↔ (𝑢 ∩ {𝑧}) ≠ ∅))
52	51	rabbidv 3419	. . . . . . 7 ⊢ (𝑦 = {𝑧} → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} = {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅})
53	52	eleq1d 2845	. . . . . 6 ⊢ (𝑦 = {𝑧} → ({𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin ↔ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin))
54	49, 53	anbi12d 644	. . . . 5 ⊢ (𝑦 = {𝑧} → ((𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin) ↔ (𝑧 ∈ {𝑧} ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)))
55	54	rspcev 3576	. . . 4 ⊢ (({𝑧} ∈ 𝒫 𝑋 ∧ (𝑧 ∈ {𝑧} ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin))
56	4, 6, 48, 55	syl12anc 850	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin))
57	56	ralrimiva 3154	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin))
58		unipw 5417	. . . 4 ⊢ ∪ 𝒫 𝑋 = 𝑋
59	58	eqcomi 2769	. . 3 ⊢ 𝑋 = ∪ 𝒫 𝑋
60	10	unisngl 23846	. . 3 ⊢ 𝑋 = ∪ 𝐶
61	59, 60	islocfin 23836	. 2 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑋 ∧ ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin)))
62	1, 2, 57, 61	syl3anbrc 1362	1 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ∈ (LocFin‘𝒫 𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ∪ cuni 4866  ‘cfv 6538  Fincfn 8973  Topctop 23211  LocFinclocfin 23823

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-en 8974  df-fin 8977  df-top 23212  df-locfin 23826

This theorem is used by:  dispcmp  34491