Theorem dispcmp 30467

Description: Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
dispcmp	⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp)

Proof of Theorem dispcmp

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

Step	Hyp	Ref	Expression
1		distop 21177	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
2		simpr 479	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
3		snelpwi 5135	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
4	3	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
5	2, 4	eqeltrd 2906	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
6	5	rexlimiva 3237	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
7	6	abssi 3904	. . . . . . . . 9 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋
8		simpl 476	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → 𝑢 = 𝑣)
9		simpr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
10	9	sneqd 4411	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → {𝑥} = {𝑧})
11	8, 10	eqeq12d 2840	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
12	11	cbvrexdva 3390	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}))
13	12	cbvabv 2952	. . . . . . . . . . 11 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}}
14	13	dissnlocfin 21710	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15		elpwg 4388	. . . . . . . . . 10 ⊢ ({𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋) → ({𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋 ↔ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ({𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋 ↔ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋))
17	7, 16	mpbiri 250	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
18	17	ad2antrr 717	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
19	14	ad2antrr 717	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
20	18, 19	elind 4027	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21		simpll 783	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → 𝑋 ∈ 𝑉)
22		simpr 479	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → 𝑋 = ∪ 𝑦)
23	22	eqcomd 2831	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → ∪ 𝑦 = 𝑋)
24	13	dissnref 21709	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
25	21, 23, 24	syl2anc 579	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
26		breq1 4878	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦))
27	26	rspcev 3526	. . . . . 6 ⊢ (({𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)) ∧ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
28	20, 25, 27	syl2anc 579	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
29	28	ex 403	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
30	29	ralrimiva 3175	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31		unipw 5141	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
32	31	eqcomi 2834	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
33	32	iscref 30452	. . 3 ⊢ (𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)))
34	1, 30, 33	sylanbrc 578	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35		ispcmp 30465	. 2 ⊢ (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
36	34, 35	sylibr 226	1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  {cab 2811  ∀wral 3117  ∃wrex 3118   ∩ cin 3797   ⊆ wss 3798  𝒫 cpw 4380  {csn 4399  ∪ cuni 4660   class class class wbr 4875  ‘cfv 6127  Topctop 21075  Refcref 21683  LocFinclocfin 21685  CovHasRefccref 30450  Paracompcpcmp 30463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-1o 7831  df-en 8229  df-fin 8232  df-top 21076  df-ref 21686  df-locfin 21688  df-cref 30451  df-pcmp 30464

This theorem is referenced by: (None)