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Theorem discmp 21249
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 20847 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2 pwfi 8302 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32biimpi 206 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
41, 3elind 3831 . . 3 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5 fincmp 21244 . . 3 (𝒫 𝐴 ∈ (Top ∩ Fin) → 𝒫 𝐴 ∈ Comp)
64, 5syl 17 . 2 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Comp)
7 simpr 476 . . . . . . . 8 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → 𝑥𝐴)
87snssd 4372 . . . . . . 7 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ⊆ 𝐴)
9 snex 4938 . . . . . . . 8 {𝑥} ∈ V
109elpw 4197 . . . . . . 7 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
118, 10sylibr 224 . . . . . 6 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ∈ 𝒫 𝐴)
12 eqid 2651 . . . . . 6 (𝑥𝐴 ↦ {𝑥}) = (𝑥𝐴 ↦ {𝑥})
1311, 12fmptd 6425 . . . . 5 (𝒫 𝐴 ∈ Comp → (𝑥𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
14 frn 6091 . . . . 5 ((𝑥𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴 → ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
1513, 14syl 17 . . . 4 (𝒫 𝐴 ∈ Comp → ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
1612rnmpt 5403 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
1716unieqi 4477 . . . . . 6 ran (𝑥𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
189dfiun2 4586 . . . . . 6 𝑥𝐴 {𝑥} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
19 iunid 4607 . . . . . 6 𝑥𝐴 {𝑥} = 𝐴
2017, 18, 193eqtr2ri 2680 . . . . 5 𝐴 = ran (𝑥𝐴 ↦ {𝑥})
2120a1i 11 . . . 4 (𝒫 𝐴 ∈ Comp → 𝐴 = ran (𝑥𝐴 ↦ {𝑥}))
22 unipw 4948 . . . . . 6 𝒫 𝐴 = 𝐴
2322eqcomi 2660 . . . . 5 𝐴 = 𝒫 𝐴
2423cmpcov 21240 . . . 4 ((𝒫 𝐴 ∈ Comp ∧ ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴𝐴 = ran (𝑥𝐴 ↦ {𝑥})) → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
2515, 21, 24mpd3an23 1466 . . 3 (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
26 elin 3829 . . . . . . 7 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) ↔ (𝑦 ∈ 𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∧ 𝑦 ∈ Fin))
2726simprbi 479 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
2826simplbi 475 . . . . . . . 8 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥𝐴 ↦ {𝑥}))
2928elpwid 4203 . . . . . . 7 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥𝐴 ↦ {𝑥}))
30 snfi 8079 . . . . . . . . . 10 {𝑥} ∈ Fin
3130rgenw 2953 . . . . . . . . 9 𝑥𝐴 {𝑥} ∈ Fin
3212fmpt 6421 . . . . . . . . 9 (∀𝑥𝐴 {𝑥} ∈ Fin ↔ (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin)
3331, 32mpbi 220 . . . . . . . 8 (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin
34 frn 6091 . . . . . . . 8 ((𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin → ran (𝑥𝐴 ↦ {𝑥}) ⊆ Fin)
3533, 34mp1i 13 . . . . . . 7 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → ran (𝑥𝐴 ↦ {𝑥}) ⊆ Fin)
3629, 35sstrd 3646 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
37 unifi 8296 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝑦 ⊆ Fin) → 𝑦 ∈ Fin)
3827, 36, 37syl2anc 694 . . . . 5 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
39 eleq1 2718 . . . . 5 (𝐴 = 𝑦 → (𝐴 ∈ Fin ↔ 𝑦 ∈ Fin))
4038, 39syl5ibrcom 237 . . . 4 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → (𝐴 = 𝑦𝐴 ∈ Fin))
4140rexlimiv 3056 . . 3 (∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦𝐴 ∈ Fin)
4225, 41syl 17 . 2 (𝒫 𝐴 ∈ Comp → 𝐴 ∈ Fin)
436, 42impbii 199 1 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552  cmpt 4762  ran crn 5144  wf 5922  Fincfn 7997  Topctop 20746  Compccmp 21237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-top 20747  df-cmp 21238
This theorem is referenced by:  disllycmp  21349  xkohaus  21504  xkoptsub  21505  xkopt  21506
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