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Theorem dispcmp 33303
Description: Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Assertion
Ref Expression
dispcmp (𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)

Proof of Theorem dispcmp
Dummy variables 𝑣 𝑦 𝑧 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 22818 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2 simpr 484 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 = {𝑥})
3 snelpwi 5443 . . . . . . . . . . . . 13 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
43adantr 480 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
52, 4eqeltrd 2832 . . . . . . . . . . 11 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
65rexlimiva 3146 . . . . . . . . . 10 (∃𝑥𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
76abssi 4067 . . . . . . . . 9 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋
8 simpl 482 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑢 = 𝑣)
9 simpr 484 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑥 = 𝑧)
109sneqd 4640 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → {𝑥} = {𝑧})
118, 10eqeq12d 2747 . . . . . . . . . . . . 13 ((𝑢 = 𝑣𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
1211cbvrexdva 3236 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (∃𝑥𝑋 𝑢 = {𝑥} ↔ ∃𝑧𝑋 𝑣 = {𝑧}))
1312cbvabv 2804 . . . . . . . . . . 11 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧𝑋 𝑣 = {𝑧}}
1413dissnlocfin 23353 . . . . . . . . . 10 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15 elpwg 4605 . . . . . . . . . 10 ({𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋) → ({𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋 ↔ {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋))
1614, 15syl 17 . . . . . . . . 9 (𝑋𝑉 → ({𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋 ↔ {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋))
177, 16mpbiri 258 . . . . . . . 8 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1817ad2antrr 723 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1914ad2antrr 723 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
2018, 19elind 4194 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21 simpll 764 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋𝑉)
22 simpr 484 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋 = 𝑦)
2322eqcomd 2737 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑦 = 𝑋)
2413dissnref 23352 . . . . . . 7 ((𝑋𝑉 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
2521, 23, 24syl2anc 583 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
26 breq1 5151 . . . . . . 7 (𝑧 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦))
2726rspcev 3612 . . . . . 6 (({𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)) ∧ {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
2820, 25, 27syl2anc 583 . . . . 5 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
2928ex 412 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
3029ralrimiva 3145 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31 unipw 5450 . . . . 5 𝒫 𝑋 = 𝑋
3231eqcomi 2740 . . . 4 𝑋 = 𝒫 𝑋
3332iscref 33288 . . 3 (𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)))
341, 30, 33sylanbrc 582 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35 ispcmp 33301 . 2 (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
3634, 35sylibr 233 1 (𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  {cab 2708  wral 3060  wrex 3069  cin 3947  wss 3948  𝒫 cpw 4602  {csn 4628   cuni 4908   class class class wbr 5148  cfv 6543  Topctop 22715  Refcref 23326  LocFinclocfin 23328  CovHasRefccref 33286  Paracompcpcmp 33299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7860  df-1o 8472  df-en 8946  df-fin 8949  df-top 22716  df-ref 23329  df-locfin 23331  df-cref 33287  df-pcmp 33300
This theorem is referenced by: (None)
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