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Theorem discmp 23422
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 23018 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2 pwfi 9355 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32biimpi 216 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
41, 3elind 4210 . . 3 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5 fincmp 23417 . . 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 4814 . . . . . . 7 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ⊆ 𝐴)
9 vsnex 5440 . . . . . . . 8 {𝑥} ∈ V
109elpw 4609 . . . . . . 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 6745 . . . 4 (𝒫 𝐴 ∈ Comp → ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14 eqid 2735 . . . . . . . 8 (𝑥𝐴 ↦ {𝑥}) = (𝑥𝐴 ↦ {𝑥})
1514rnmpt 5971 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
1615unieqi 4924 . . . . . 6 ran (𝑥𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
179dfiun2 5038 . . . . . 6 𝑥𝐴 {𝑥} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
18 iunid 5065 . . . . . 6 𝑥𝐴 {𝑥} = 𝐴
1916, 17, 183eqtr2ri 2770 . . . . 5 𝐴 = ran (𝑥𝐴 ↦ {𝑥})
2019a1i 11 . . . 4 (𝒫 𝐴 ∈ Comp → 𝐴 = ran (𝑥𝐴 ↦ {𝑥}))
21 unipw 5461 . . . . . 6 𝒫 𝐴 = 𝐴
2221eqcomi 2744 . . . . 5 𝐴 = 𝒫 𝐴
2322cmpcov 23413 . . . 4 ((𝒫 𝐴 ∈ Comp ∧ ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴𝐴 = ran (𝑥𝐴 ↦ {𝑥})) → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
2413, 20, 23mpd3an23 1462 . . 3 (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
25 elinel2 4212 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26 elinel1 4211 . . . . . . . 8 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥𝐴 ↦ {𝑥}))
2726elpwid 4614 . . . . . . 7 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥𝐴 ↦ {𝑥}))
28 snfi 9082 . . . . . . . . . 10 {𝑥} ∈ Fin
2928rgenw 3063 . . . . . . . . 9 𝑥𝐴 {𝑥} ∈ Fin
3014fmpt 7130 . . . . . . . . 9 (∀𝑥𝐴 {𝑥} ∈ Fin ↔ (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin)
3129, 30mpbi 230 . . . . . . . 8 (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin
32 frn 6744 . . . . . . . 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 4006 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35 unifi 9382 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝑦 ⊆ Fin) → 𝑦 ∈ Fin)
3625, 34, 35syl2anc 584 . . . . 5 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
37 eleq1 2827 . . . . 5 (𝐴 = 𝑦 → (𝐴 ∈ Fin ↔ 𝑦 ∈ Fin))
3836, 37syl5ibrcom 247 . . . 4 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → (𝐴 = 𝑦𝐴 ∈ Fin))
3938rexlimiv 3146 . . 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 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  cin 3962  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912   ciun 4996  cmpt 5231  ran crn 5690  wf 6559  Fincfn 8984  Topctop 22915  Compccmp 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988  df-top 22916  df-cmp 23411
This theorem is referenced by:  disllycmp  23522  xkohaus  23677  xkoptsub  23678  xkopt  23679
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