Theorem dispcmp 34019

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 22970	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
2		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
3		snelpwi 5391	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
4	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
5	2, 4	eqeltrd 2837	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
6	5	rexlimiva 3131	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
7	6	abssi 4009	. . . . . . . . 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 4580	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → {𝑥} = {𝑧})
11	8, 10	eqeq12d 2753	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
12	11	cbvrexdva 3219	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}))
13	12	cbvabv 2807	. . . . . . . . . . 11 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}}
14	13	dissnlocfin 23504	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15		elpwg 4545	. . . . . . . . . 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 727	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
19	14	ad2antrr 727	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
20	18, 19	elind 4141	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21		simpll 767	. . . . . . 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 2743	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → ∪ 𝑦 = 𝑋)
24	13	dissnref 23503	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
25	21, 23, 24	syl2anc 585	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
26		breq1 5089	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦))
27	26	rspcev 3565	. . . . . 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 585	. . . . 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 3130	. . 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 2746	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
33	32	iscref 34004	. . 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 584	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35		ispcmp 34017	. 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 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851   class class class wbr 5086  ‘cfv 6492  Topctop 22868  Refcref 23477  LocFinclocfin 23479  CovHasRefccref 34002  Paracompcpcmp 34015

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 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7811  df-1o 8398  df-en 8887  df-fin 8890  df-top 22869  df-ref 23480  df-locfin 23482  df-cref 34003  df-pcmp 34016

This theorem is referenced by: (None)