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Theorem dispcmp 31116
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 21595 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2 simpr 487 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 = {𝑥})
3 snelpwi 5327 . . . . . . . . . . . . 13 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
43adantr 483 . . . . . . . . . . . 12 ((𝑥𝑋𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
52, 4eqeltrd 2911 . . . . . . . . . . 11 ((𝑥𝑋𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
65rexlimiva 3279 . . . . . . . . . 10 (∃𝑥𝑋 𝑢 = {𝑥} → 𝑢 ∈ 𝒫 𝑋)
76abssi 4044 . . . . . . . . 9 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ⊆ 𝒫 𝑋
8 simpl 485 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑢 = 𝑣)
9 simpr 487 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑣𝑥 = 𝑧) → 𝑥 = 𝑧)
109sneqd 4571 . . . . . . . . . . . . . 14 ((𝑢 = 𝑣𝑥 = 𝑧) → {𝑥} = {𝑧})
118, 10eqeq12d 2835 . . . . . . . . . . . . 13 ((𝑢 = 𝑣𝑥 = 𝑧) → (𝑢 = {𝑥} ↔ 𝑣 = {𝑧}))
1211cbvrexdva 3459 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (∃𝑥𝑋 𝑢 = {𝑥} ↔ ∃𝑧𝑋 𝑣 = {𝑧}))
1312cbvabv 2887 . . . . . . . . . . 11 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = {𝑣 ∣ ∃𝑧𝑋 𝑣 = {𝑧}}
1413dissnlocfin 22129 . . . . . . . . . 10 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
15 elpwg 4543 . . . . . . . . . 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 260 . . . . . . . 8 (𝑋𝑉 → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1817ad2antrr 724 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ 𝒫 𝒫 𝑋)
1914ad2antrr 724 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (LocFin‘𝒫 𝑋))
2018, 19elind 4169 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋)))
21 simpll 765 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋𝑉)
22 simpr 487 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑋 = 𝑦)
2322eqcomd 2825 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → 𝑦 = 𝑋)
2413dissnref 22128 . . . . . . 7 ((𝑋𝑉 𝑦 = 𝑋) → {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}Ref𝑦)
2521, 23, 24syl2anc 586 . . . . . 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 3621 . . . . . 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 586 . . . . 5 (((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) ∧ 𝑋 = 𝑦) → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦)
2928ex 415 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
3029ralrimiva 3180 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝒫 𝑋 ∩ (LocFin‘𝒫 𝑋))𝑧Ref𝑦))
31 unipw 5333 . . . . 5 𝒫 𝑋 = 𝑋
3231eqcomi 2828 . . . 4 𝑋 = 𝒫 𝑋
3332iscref 31101 . . 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 585 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
35 ispcmp 31114 . 2 (𝒫 𝑋 ∈ Paracomp ↔ 𝒫 𝑋 ∈ CovHasRef(LocFin‘𝒫 𝑋))
3634, 35sylibr 236 1 (𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  {cab 2797  wral 3136  wrex 3137  cin 3933  wss 3934  𝒫 cpw 4537  {csn 4559   cuni 4830   class class class wbr 5057  cfv 6348  Topctop 21493  Refcref 22102  LocFinclocfin 22104  CovHasRefccref 31099  Paracompcpcmp 31112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  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 7573  df-1o 8094  df-en 8502  df-fin 8505  df-top 21494  df-ref 22105  df-locfin 22107  df-cref 31100  df-pcmp 31113
This theorem is referenced by: (None)
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