Theorem dispcmp 31711

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 22053	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top)
2		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
3		snelpwi 5354	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
4	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
5	2, 4	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
6	5	rexlimiva 3209	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
7	6	abssi 3999	. . . . . . . . 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 4570	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → {𝑥} = {𝑧})
11	8, 10	eqeq12d 2754	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑣 ∧ 𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
12	11	cbvrexdva 3384	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (∃𝑥 ∈ 𝑋 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}))
13	12	cbvabv 2812	. . . . . . . . . . 11 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧 ∈ 𝑋 𝑣 = {𝑧}}
14	13	dissnlocfin 22588	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15		elpwg 4533	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
18	17	ad2antrr 722	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
19	14	ad2antrr 722	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
20	18, 19	elind 4124	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21		simpll 763	. . . . . . 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 2744	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → ∪ 𝑦 = 𝑋)
24	13	dissnref 22587	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
25	21, 23, 24	syl2anc 583	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = ∪ 𝑦) → {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦)
26		breq1 5073	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}Ref𝑦))
27	26	rspcev 3552	. . . . . 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 583	. . . . 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 3107	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31		unipw 5360	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
32	31	eqcomi 2747	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
33	32	iscref 31696	. . 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 582	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35		ispcmp 31709	. 2 ⊢ (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
36	34, 35	sylibr 233	1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   class class class wbr 5070  ‘cfv 6418  Topctop 21950  Refcref 22561  LocFinclocfin 22563  CovHasRefccref 31694  Paracompcpcmp 31707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-en 8692  df-fin 8695  df-top 21951  df-ref 22564  df-locfin 22566  df-cref 31695  df-pcmp 31708

This theorem is referenced by: (None)