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Theorem discmp 21934
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 21531 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2 pwfi 8807 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32biimpi 217 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
41, 3elind 4168 . . 3 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5 fincmp 21929 . . 3 (𝒫 𝐴 ∈ (Top ∩ Fin) → 𝒫 𝐴 ∈ Comp)
64, 5syl 17 . 2 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Comp)
7 simpr 485 . . . . . . . 8 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → 𝑥𝐴)
87snssd 4734 . . . . . . 7 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ⊆ 𝐴)
9 snex 5322 . . . . . . . 8 {𝑥} ∈ V
109elpw 4542 . . . . . . 7 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
118, 10sylibr 235 . . . . . 6 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ∈ 𝒫 𝐴)
1211fmpttd 6871 . . . . 5 (𝒫 𝐴 ∈ Comp → (𝑥𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
1312frnd 6514 . . . 4 (𝒫 𝐴 ∈ Comp → ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14 eqid 2818 . . . . . . . 8 (𝑥𝐴 ↦ {𝑥}) = (𝑥𝐴 ↦ {𝑥})
1514rnmpt 5820 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
1615unieqi 4839 . . . . . 6 ran (𝑥𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
179dfiun2 4949 . . . . . 6 𝑥𝐴 {𝑥} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
18 iunid 4975 . . . . . 6 𝑥𝐴 {𝑥} = 𝐴
1916, 17, 183eqtr2ri 2848 . . . . 5 𝐴 = ran (𝑥𝐴 ↦ {𝑥})
2019a1i 11 . . . 4 (𝒫 𝐴 ∈ Comp → 𝐴 = ran (𝑥𝐴 ↦ {𝑥}))
21 unipw 5333 . . . . . 6 𝒫 𝐴 = 𝐴
2221eqcomi 2827 . . . . 5 𝐴 = 𝒫 𝐴
2322cmpcov 21925 . . . 4 ((𝒫 𝐴 ∈ Comp ∧ ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴𝐴 = ran (𝑥𝐴 ↦ {𝑥})) → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
2413, 20, 23mpd3an23 1454 . . 3 (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
25 elinel2 4170 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26 elinel1 4169 . . . . . . . 8 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥𝐴 ↦ {𝑥}))
2726elpwid 4549 . . . . . . 7 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥𝐴 ↦ {𝑥}))
28 snfi 8582 . . . . . . . . . 10 {𝑥} ∈ Fin
2928rgenw 3147 . . . . . . . . 9 𝑥𝐴 {𝑥} ∈ Fin
3014fmpt 6866 . . . . . . . . 9 (∀𝑥𝐴 {𝑥} ∈ Fin ↔ (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin)
3129, 30mpbi 231 . . . . . . . 8 (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin
32 frn 6513 . . . . . . . 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 3974 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35 unifi 8801 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝑦 ⊆ Fin) → 𝑦 ∈ Fin)
3625, 34, 35syl2anc 584 . . . . 5 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
37 eleq1 2897 . . . . 5 (𝐴 = 𝑦 → (𝐴 ∈ Fin ↔ 𝑦 ∈ Fin))
3836, 37syl5ibrcom 248 . . . 4 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → (𝐴 = 𝑦𝐴 ∈ Fin))
3938rexlimiv 3277 . . 3 (∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦𝐴 ∈ Fin)
4024, 39syl 17 . 2 (𝒫 𝐴 ∈ Comp → 𝐴 ∈ Fin)
416, 40impbii 210 1 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  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   ciun 4910  cmpt 5137  ran crn 5549  wf 6344  Fincfn 8497  Topctop 21429  Compccmp 21922
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-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  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-int 4868  df-iun 4912  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-pred 6141  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-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-top 21430  df-cmp 21923
This theorem is referenced by:  disllycmp  22034  xkohaus  22189  xkoptsub  22190  xkopt  22191
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