Theorem dispcmp 33825

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 22898	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
2		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
3		snelpwi 5390	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
4	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
5	2, 4	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
6	5	rexlimiva 3122	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
7	6	abssi 4023	. . . . . . . . 9 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋
8		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → 𝑢 = 𝑣)
9		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
10	9	sneqd 4591	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → {𝑥} = {𝑧})
11	8, 10	eqeq12d 2745	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
12	11	cbvrexdva 3210	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}))
13	12	cbvabv 2799	. . . . . . . . . . 11 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}}
14	13	dissnlocfin 23432	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15		elpwg 4556	. . . . . . . . . 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 258	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
18	17	ad2antrr 726	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
19	14	ad2antrr 726	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
20	18, 19	elind 4153	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21		simpll 766	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → 𝑋 ∈ 𝑉)
22		simpr 484	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → 𝑋 = ∪ 𝑦)
23	22	eqcomd 2735	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → ∪ 𝑦 = 𝑋)
24	13	dissnref 23431	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
25	21, 23, 24	syl2anc 584	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
26		breq1 5098	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦))
27	26	rspcev 3579	. . . . . 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 584	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
29	28	ex 412	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
30	29	ralrimiva 3121	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31		unipw 5397	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
32	31	eqcomi 2738	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
33	32	iscref 33810	. . 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 583	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35		ispcmp 33823	. 2 ⊢ (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
36	34, 35	sylibr 234	1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553  {csn 4579  ∪ cuni 4861   class class class wbr 5095  ‘cfv 6486  Topctop 22796  Refcref 23405  LocFinclocfin 23407  CovHasRefccref 33808  Paracompcpcmp 33821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7807  df-1o 8395  df-en 8880  df-fin 8883  df-top 22797  df-ref 23408  df-locfin 23410  df-cref 33809  df-pcmp 33822

This theorem is referenced by: (None)