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Theorem dispcmp 31022
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 21531 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2 simpr 485 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 = {𝑥})
3 snelpwi 5327 . . . . . . . . . . . . 13 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
43adantr 481 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
52, 4eqeltrd 2910 . . . . . . . . . . 11 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
65rexlimiva 3278 . . . . . . . . . 10 (∃𝑥𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
76abssi 4043 . . . . . . . . 9 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋
8 simpl 483 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑢 = 𝑣)
9 simpr 485 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑥 = 𝑧)
109sneqd 4569 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → {𝑥} = {𝑧})
118, 10eqeq12d 2834 . . . . . . . . . . . . 13 ((𝑢 = 𝑣𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
1211cbvrexdva 3458 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (∃𝑥𝑋 𝑢 = {𝑥} ↔ ∃𝑧𝑋 𝑣 = {𝑧}))
1312cbvabv 2886 . . . . . . . . . . 11 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧𝑋 𝑣 = {𝑧}}
1413dissnlocfin 22065 . . . . . . . . . 10 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15 elpwg 4541 . . . . . . . . . 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 259 . . . . . . . 8 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1817ad2antrr 722 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1914ad2antrr 722 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
2018, 19elind 4168 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21 simpll 763 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋𝑉)
22 simpr 485 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋 = 𝑦)
2322eqcomd 2824 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑦 = 𝑋)
2413dissnref 22064 . . . . . . 7 ((𝑋𝑉 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
2521, 23, 24syl2anc 584 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
26 breq1 5060 . . . . . . 7 (𝑧 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦))
2726rspcev 3620 . . . . . 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 584 . . . . 5 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
2928ex 413 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
3029ralrimiva 3179 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31 unipw 5333 . . . . 5 𝒫 𝑋 = 𝑋
3231eqcomi 2827 . . . 4 𝑋 = 𝒫 𝑋
3332iscref 31007 . . 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 583 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35 ispcmp 31020 . 2 (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
3634, 35sylibr 235 1 (𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wrex 3136  cin 3932  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830   class class class wbr 5057  cfv 6348  Topctop 21429  Refcref 22038  LocFinclocfin 22040  CovHasRefccref 31005  Paracompcpcmp 31018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091  df-en 8498  df-fin 8501  df-top 21430  df-ref 22041  df-locfin 22043  df-cref 31006  df-pcmp 31019
This theorem is referenced by: (None)
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