Theorem dispcmp 34602

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 23368	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
2		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
3		snelpwi 5419	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
4	3	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
5	2, 4	eqeltrd 2864	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
6	5	rexlimiva 3159	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
7	6	abssi 3996	. . . . . . . . 9 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋
8		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → 𝑢 = 𝑣)
9		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
10	9	sneqd 4596	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → {𝑥} = {𝑧})
11	8, 10	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
12	11	cbvrexdva 3247	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}))
13	12	cbvabv 2834	. . . . . . . . . . 11 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}}
14	13	dissnlocfin 23903	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15		elpwg 4563	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ({𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋 ↔ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋))
17	7, 16	mpbiri 261	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
18	17	ad2antrr 739	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
19	14	ad2antrr 739	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
20	18, 19	elind 4100	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21		simpll 779	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → 𝑋 ∈ 𝑉)
22		simpr 490	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → 𝑋 = ∪ 𝑦)
23	22	eqcomd 2769	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → ∪ 𝑦 = 𝑋)
24	13	dissnref 23902	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
25	21, 23, 24	syl2anc 596	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
26		breq1 5111	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦))
27	26	rspcev 3580	. . . . . 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 596	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
29	28	ex 418	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
30	29	ralrimiva 3158	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31		unipw 5425	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
32	31	eqcomi 2772	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
33	32	iscref 34587	. . 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 595	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35		ispcmp 34600	. 2 ⊢ (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
36	34, 35	sylibr 237	1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  {csn 4588  ∪ cuni 4872   class class class wbr 5108  ‘cfv 6566  Topctop 23266  Refcref 23876  LocFinclocfin 23878  CovHasRefccref 34585  Paracompcpcmp 34598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-om 7912  df-1o 8514  df-en 9012  df-fin 9015  df-top 23267  df-ref 23879  df-locfin 23881  df-cref 34586  df-pcmp 34599

This theorem is used by: (None)