Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dispcmp Structured version   Visualization version   GIF version

Theorem dispcmp 30251
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 21013 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2 simpr 473 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 = {𝑥})
3 snelpwi 5102 . . . . . . . . . . . . 13 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
43adantr 468 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
52, 4eqeltrd 2885 . . . . . . . . . . 11 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
65rexlimiva 3216 . . . . . . . . . 10 (∃𝑥𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
76abssi 3874 . . . . . . . . 9 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋
8 simpl 470 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑢 = 𝑣)
9 simpr 473 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑥 = 𝑧)
109sneqd 4382 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → {𝑥} = {𝑧})
118, 10eqeq12d 2821 . . . . . . . . . . . . 13 ((𝑢 = 𝑣𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
1211cbvrexdva 3367 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (∃𝑥𝑋 𝑢 = {𝑥} ↔ ∃𝑧𝑋 𝑣 = {𝑧}))
1312cbvabv 2931 . . . . . . . . . . 11 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧𝑋 𝑣 = {𝑧}}
1413dissnlocfin 21546 . . . . . . . . . 10 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15 elpwg 4359 . . . . . . . . . 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 249 . . . . . . . 8 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1817ad2antrr 708 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1914ad2antrr 708 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
2018, 19elind 3997 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21 simpll 774 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋𝑉)
22 simpr 473 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋 = 𝑦)
2322eqcomd 2812 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑦 = 𝑋)
2413dissnref 21545 . . . . . . 7 ((𝑋𝑉 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
2521, 23, 24syl2anc 575 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
26 breq1 4847 . . . . . . 7 (𝑧 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} → (𝑧Ref𝑦 ↔ {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦))
2726rspcev 3502 . . . . . 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 575 . . . . 5 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
2928ex 399 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
3029ralrimiva 3154 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31 unipw 5108 . . . . 5 𝒫 𝑋 = 𝑋
3231eqcomi 2815 . . . 4 𝑋 = 𝒫 𝑋
3332iscref 30236 . . 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 574 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35 ispcmp 30249 . 2 (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
3634, 35sylibr 225 1 (𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  {cab 2792  wral 3096  wrex 3097  cin 3768  wss 3769  𝒫 cpw 4351  {csn 4370   cuni 4630   class class class wbr 4844  cfv 6101  Topctop 20911  Refcref 21519  LocFinclocfin 21521  CovHasRefccref 30234  Paracompcpcmp 30247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7296  df-1o 7796  df-en 8193  df-fin 8196  df-top 20912  df-ref 21522  df-locfin 21524  df-cref 30235  df-pcmp 30248
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator