Theorem dissnlocfin 21332

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 20799	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
2		eqidd 2623	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 = 𝑋)
3		snelpwi 4912	. . . . 5 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ 𝒫 𝑋)
4	3	adantl 482	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑧} ∈ 𝒫 𝑋)
5		vsnid 4209	. . . . 5 ⊢ 𝑧 ∈ {𝑧}
6	5	a1i 11	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ {𝑧})
7		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑢(𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋)
8		nfrab1 3122	. . . . . 6 ⊢ Ⅎ𝑢{𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅}
9		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑢{{𝑧}}
10		dissnref.c	. . . . . . . . . 10 ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}
11	10	abeq2i 2735	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
12	11	anbi1i 731	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
13		simpr 477	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
14		simplr 792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → 𝑢 = {𝑥})
15	14	ineq1d 3813	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) = ({𝑥} ∩ {𝑧}))
16		disjsn2 4247	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ≠ 𝑧 → ({𝑥} ∩ {𝑧}) = ∅)
17	16	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → ({𝑥} ∩ {𝑧}) = ∅)
18	15, 17	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) = ∅)
19		simp-4r 807	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → (𝑢 ∩ {𝑧}) ≠ ∅)
20	19	neneqd 2799	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ 𝑥 ≠ 𝑧) → ¬ (𝑢 ∩ {𝑧}) = ∅)
21	18, 20	pm2.65da 600	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → ¬ 𝑥 ≠ 𝑧)
22		nne 2798	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ≠ 𝑧 ↔ 𝑥 = 𝑧)
23	21, 22	sylib 208	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 = 𝑧)
24	23	sneqd 4189	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} = {𝑧})
25	13, 24	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑧})
26	25	r19.29an 3077	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}) → 𝑢 = {𝑧})
27	26	an32s 846	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}) ∧ (𝑢 ∩ {𝑧}) ≠ ∅) → 𝑢 = {𝑧})
28	27	anasss 679	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅)) → 𝑢 = {𝑧})
29		sneq 4187	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
30	29	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
31	30	rspcev 3309	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑢 = {𝑧}) → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
32	31	adantll 750	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
33		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → 𝑢 = {𝑧})
34	33	ineq1d 3813	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) = ({𝑧} ∩ {𝑧}))
35		inidm 3822	. . . . . . . . . . . 12 ⊢ ({𝑧} ∩ {𝑧}) = {𝑧}
36	34, 35	syl6eq 2672	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) = {𝑧})
37		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
38	37	snnz 4309	. . . . . . . . . . . 12 ⊢ {𝑧} ≠ ∅
39	38	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → {𝑧} ≠ ∅)
40	36, 39	eqnetrd 2861	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (𝑢 ∩ {𝑧}) ≠ ∅)
41	32, 40	jca 554	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ 𝑢 = {𝑧}) → (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
42	28, 41	impbida 877	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ((∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ 𝑢 = {𝑧}))
43	12, 42	syl5bb 272	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ((𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅) ↔ 𝑢 = {𝑧}))
44		rabid 3116	. . . . . . 7 ⊢ (𝑢 ∈ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ↔ (𝑢 ∈ 𝐶 ∧ (𝑢 ∩ {𝑧}) ≠ ∅))
45		velsn 4193	. . . . . . 7 ⊢ (𝑢 ∈ {{𝑧}} ↔ 𝑢 = {𝑧})
46	43, 44, 45	3bitr4g 303	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → (𝑢 ∈ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ↔ 𝑢 ∈ {{𝑧}}))
47	7, 8, 9, 46	eqrd 3622	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} = {{𝑧}})
48		snfi 8038	. . . . 5 ⊢ {{𝑧}} ∈ Fin
49	47, 48	syl6eqel 2709	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)
50		eleq2 2690	. . . . . 6 ⊢ (𝑦 = {𝑧} → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ {𝑧}))
51		ineq2 3808	. . . . . . . . 9 ⊢ (𝑦 = {𝑧} → (𝑢 ∩ 𝑦) = (𝑢 ∩ {𝑧}))
52	51	neeq1d 2853	. . . . . . . 8 ⊢ (𝑦 = {𝑧} → ((𝑢 ∩ 𝑦) ≠ ∅ ↔ (𝑢 ∩ {𝑧}) ≠ ∅))
53	52	rabbidv 3189	. . . . . . 7 ⊢ (𝑦 = {𝑧} → {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} = {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅})
54	53	eleq1d 2686	. . . . . 6 ⊢ (𝑦 = {𝑧} → ({𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin ↔ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin))
55	50, 54	anbi12d 747	. . . . 5 ⊢ (𝑦 = {𝑧} → ((𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin) ↔ (𝑧 ∈ {𝑧} ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)))
56	55	rspcev 3309	. . . 4 ⊢ (({𝑧} ∈ 𝒫 𝑋 ∧ (𝑧 ∈ {𝑧} ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ {𝑧}) ≠ ∅} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin))
57	4, 6, 49, 56	syl12anc 1324	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin))
58	57	ralrimiva 2966	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin))
59		unipw 4918	. . . 4 ⊢ ∪ 𝒫 𝑋 = 𝑋
60	59	eqcomi 2631	. . 3 ⊢ 𝑋 = ∪ 𝒫 𝑋
61	10	unisngl 21330	. . 3 ⊢ 𝑋 = ∪ 𝐶
62	60, 61	islocfin 21320	. 2 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑋 ∧ ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑧 ∈ 𝑦 ∧ {𝑢 ∈ 𝐶 ∣ (𝑢 ∩ 𝑦) ≠ ∅} ∈ Fin)))
63	1, 2, 58, 62	syl3anbrc 1246	1 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ∈ (LocFin‘𝒫 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ∩ cin 3573  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ‘cfv 5888  Fincfn 7955  Topctop 20698  LocFinclocfin 21307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-en 7956  df-fin 7959  df-top 20699  df-locfin 21310

This theorem is referenced by:  dispcmp  29926