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Theorem discmp 23406
Description: A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
discmp (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)

Proof of Theorem discmp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 23002 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2 pwfi 9357 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32biimpi 216 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
41, 3elind 4200 . . 3 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5 fincmp 23401 . . 3 (𝒫 𝐴 ∈ (Top ∩ Fin) → 𝒫 𝐴 ∈ Comp)
64, 5syl 17 . 2 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Comp)
7 simpr 484 . . . . . . . 8 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → 𝑥𝐴)
87snssd 4809 . . . . . . 7 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ⊆ 𝐴)
9 vsnex 5434 . . . . . . . 8 {𝑥} ∈ V
109elpw 4604 . . . . . . 7 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
118, 10sylibr 234 . . . . . 6 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ∈ 𝒫 𝐴)
1211fmpttd 7135 . . . . 5 (𝒫 𝐴 ∈ Comp → (𝑥𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
1312frnd 6744 . . . 4 (𝒫 𝐴 ∈ Comp → ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14 eqid 2737 . . . . . . . 8 (𝑥𝐴 ↦ {𝑥}) = (𝑥𝐴 ↦ {𝑥})
1514rnmpt 5968 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
1615unieqi 4919 . . . . . 6 ran (𝑥𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
179dfiun2 5033 . . . . . 6 𝑥𝐴 {𝑥} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
18 iunid 5060 . . . . . 6 𝑥𝐴 {𝑥} = 𝐴
1916, 17, 183eqtr2ri 2772 . . . . 5 𝐴 = ran (𝑥𝐴 ↦ {𝑥})
2019a1i 11 . . . 4 (𝒫 𝐴 ∈ Comp → 𝐴 = ran (𝑥𝐴 ↦ {𝑥}))
21 unipw 5455 . . . . . 6 𝒫 𝐴 = 𝐴
2221eqcomi 2746 . . . . 5 𝐴 = 𝒫 𝐴
2322cmpcov 23397 . . . 4 ((𝒫 𝐴 ∈ Comp ∧ ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴𝐴 = ran (𝑥𝐴 ↦ {𝑥})) → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
2413, 20, 23mpd3an23 1465 . . 3 (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
25 elinel2 4202 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26 elinel1 4201 . . . . . . . 8 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥𝐴 ↦ {𝑥}))
2726elpwid 4609 . . . . . . 7 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥𝐴 ↦ {𝑥}))
28 snfi 9083 . . . . . . . . . 10 {𝑥} ∈ Fin
2928rgenw 3065 . . . . . . . . 9 𝑥𝐴 {𝑥} ∈ Fin
3014fmpt 7130 . . . . . . . . 9 (∀𝑥𝐴 {𝑥} ∈ Fin ↔ (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin)
3129, 30mpbi 230 . . . . . . . 8 (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin
32 frn 6743 . . . . . . . 8 ((𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin → ran (𝑥𝐴 ↦ {𝑥}) ⊆ Fin)
3331, 32mp1i 13 . . . . . . 7 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → ran (𝑥𝐴 ↦ {𝑥}) ⊆ Fin)
3427, 33sstrd 3994 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35 unifi 9384 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝑦 ⊆ Fin) → 𝑦 ∈ Fin)
3625, 34, 35syl2anc 584 . . . . 5 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
37 eleq1 2829 . . . . 5 (𝐴 = 𝑦 → (𝐴 ∈ Fin ↔ 𝑦 ∈ Fin))
3836, 37syl5ibrcom 247 . . . 4 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → (𝐴 = 𝑦𝐴 ∈ Fin))
3938rexlimiv 3148 . . 3 (∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦𝐴 ∈ Fin)
4024, 39syl 17 . 2 (𝒫 𝐴 ∈ Comp → 𝐴 ∈ Fin)
416, 40impbii 209 1 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907   ciun 4991  cmpt 5225  ran crn 5686  wf 6557  Fincfn 8985  Topctop 22899  Compccmp 23394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989  df-top 22900  df-cmp 23395
This theorem is referenced by:  disllycmp  23506  xkohaus  23661  xkoptsub  23662  xkopt  23663
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