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Theorem dispcmp 30467
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 21177 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2 simpr 479 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 = {𝑥})
3 snelpwi 5135 . . . . . . . . . . . . 13 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
43adantr 474 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
52, 4eqeltrd 2906 . . . . . . . . . . 11 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
65rexlimiva 3237 . . . . . . . . . 10 (∃𝑥𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
76abssi 3904 . . . . . . . . 9 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋
8 simpl 476 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑢 = 𝑣)
9 simpr 479 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑥 = 𝑧)
109sneqd 4411 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → {𝑥} = {𝑧})
118, 10eqeq12d 2840 . . . . . . . . . . . . 13 ((𝑢 = 𝑣𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
1211cbvrexdva 3390 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (∃𝑥𝑋 𝑢 = {𝑥} ↔ ∃𝑧𝑋 𝑣 = {𝑧}))
1312cbvabv 2952 . . . . . . . . . . 11 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧𝑋 𝑣 = {𝑧}}
1413dissnlocfin 21710 . . . . . . . . . 10 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15 elpwg 4388 . . . . . . . . . 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 250 . . . . . . . 8 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1817ad2antrr 717 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1914ad2antrr 717 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
2018, 19elind 4027 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21 simpll 783 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋𝑉)
22 simpr 479 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋 = 𝑦)
2322eqcomd 2831 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑦 = 𝑋)
2413dissnref 21709 . . . . . . 7 ((𝑋𝑉 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
2521, 23, 24syl2anc 579 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
26 breq1 4878 . . . . . . 7 (𝑧 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦))
2726rspcev 3526 . . . . . 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 579 . . . . 5 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
2928ex 403 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
3029ralrimiva 3175 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31 unipw 5141 . . . . 5 𝒫 𝑋 = 𝑋
3231eqcomi 2834 . . . 4 𝑋 = 𝒫 𝑋
3332iscref 30452 . . 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 578 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35 ispcmp 30465 . 2 (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
3634, 35sylibr 226 1 (𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  {cab 2811  wral 3117  wrex 3118  cin 3797  wss 3798  𝒫 cpw 4380  {csn 4399   cuni 4660   class class class wbr 4875  cfv 6127  Topctop 21075  Refcref 21683  LocFinclocfin 21685  CovHasRefccref 30450  Paracompcpcmp 30463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-1o 7831  df-en 8229  df-fin 8232  df-top 21076  df-ref 21686  df-locfin 21688  df-cref 30451  df-pcmp 30464
This theorem is referenced by: (None)
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