Theorem dissnlocfin 23494

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 22960	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
2		eqidd 2737	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 = 𝑋)
3		snelpwi 5396	. . . . 5 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ 𝒫 𝑋)
4	3	adantl 481	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑧} ∈ 𝒫 𝑋)
5		vsnid 4607	. . . . 5 ⊢ 𝑧 ∈ {𝑧}
6	5	a1i 11	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ {𝑧})
7		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑢(𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋)
8		nfrab1 3409	. . . . . 6 ⊢ Ⅎ𝑢{𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅}
9		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑢{{𝑧}}
10		dissnref.c	. . . . . . . . . 10 ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}
11	10	eqabri 2878	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
12	11	anbi1i 625	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
13		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
14		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → 𝑢 = {𝑥})
15	14	ineq1d 4159	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) = ({𝑥} ∩ {𝑧}))
16		disjsn2 4656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ≠ 𝑧 → ({𝑥} ∩ {𝑧}) = ∅)
17	16	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → ({𝑥} ∩ {𝑧}) = ∅)
18	15, 17	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) = ∅)
19		simp-4r 784	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) ≠ ∅)
20	19	neneqd 2937	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → ¬ (𝑢 ∩ {𝑧}) = ∅)
21	18, 20	pm2.65da 817	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → ¬ 𝑥 ≠ 𝑧)
22		nne 2936	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ≠ 𝑧 ↔ 𝑥 = 𝑧)
23	21, 22	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 = 𝑧)
24	23	sneqd 4579	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} = {𝑧})
25	13, 24	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑧})
26	25	r19.29an 3141	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}) → 𝑢 = {𝑧})
27	26	an32s 653	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) → 𝑢 = {𝑧})
28	27	anasss 466	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅)) → 𝑢 = {𝑧})
29		sneq 4577	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
30	29	rspceeqv 3587	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑢 = {𝑧}) → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
31	30	adantll 715	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
32		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → 𝑢 = {𝑧})
33	32	ineq1d 4159	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) = ({𝑧} ∩ {𝑧}))
34		inidm 4167	. . . . . . . . . . . 12 ⊢ ({𝑧} ∩ {𝑧}) = {𝑧}
35	33, 34	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) = {𝑧})
36		vex 3433	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
37	36	snnz 4720	. . . . . . . . . . . 12 ⊢ {𝑧} ≠ ∅
38	37	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → {𝑧} ≠ ∅)
39	35, 38	eqnetrd 2999	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) ≠ ∅)
40	31, 39	jca 511	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
41	28, 40	impbida 801	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ((∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ 𝑢 = {𝑧}))
42	12, 41	bitrid 283	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ((𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ 𝑢 = {𝑧}))
43		rabid 3410	. . . . . . 7 ⊢ (𝑢 ∈ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ↔ (𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
44		velsn 4583	. . . . . . 7 ⊢ (𝑢 ∈ {{𝑧}} ↔ 𝑢 = {𝑧})
45	42, 43, 44	3bitr4g 314	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → (𝑢 ∈ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ↔ 𝑢 ∈ {{𝑧}}))
46	7, 8, 9, 45	eqrd 3941	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} = {{𝑧}})
47		snfi 8990	. . . . 5 ⊢ {{𝑧}} ∈ Fin
48	46, 47	eqeltrdi 2844	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)
49		eleq2 2825	. . . . . 6 ⊢ (𝑦 = {𝑧} → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ {𝑧}))
50		ineq2 4154	. . . . . . . . 9 ⊢ (𝑦 = {𝑧} → (𝑢 ∩ 𝑦) = (𝑢 ∩ {𝑧}))
51	50	neeq1d 2991	. . . . . . . 8 ⊢ (𝑦 = {𝑧} → ((𝑢 ∩ 𝑦) ≠ ∅ ↔ (𝑢 ∩ {𝑧}) ≠ ∅))
52	51	rabbidv 3396	. . . . . . 7 ⊢ (𝑦 = {𝑧} → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} = {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅})
53	52	eleq1d 2821	. . . . . 6 ⊢ (𝑦 = {𝑧} → ({𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin ↔ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin))
54	49, 53	anbi12d 633	. . . . 5 ⊢ (𝑦 = {𝑧} → ((𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin) ↔ (𝑧 ∈ {𝑧} ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)))
55	54	rspcev 3564	. . . 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 837	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin))
57	56	ralrimiva 3129	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin))
58		unipw 5402	. . . 4 ⊢ ∪ 𝒫 𝑋 = 𝑋
59	58	eqcomi 2745	. . 3 ⊢ 𝑋 = ∪ 𝒫 𝑋
60	10	unisngl 23492	. . 3 ⊢ 𝑋 = ∪ 𝐶
61	59, 60	islocfin 23482	. 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 1345	1 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ∈ (LocFin‘𝒫 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  {crab 3389   ∩ cin 3888  ∅c0 4273  𝒫 cpw 4541  {csn 4567  ∪ cuni 4850  ‘cfv 6498  Fincfn 8893  Topctop 22858  LocFinclocfin 23469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-en 8894  df-fin 8897  df-top 22859  df-locfin 23472

This theorem is referenced by:  dispcmp  34003