MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  discmp Structured version   Visualization version   GIF version

Theorem discmp 22901
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 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 22497 . . . 4 (𝐴 ∈ Fin β†’ 𝒫 𝐴 ∈ Top)
2 pwfi 9177 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32biimpi 215 . . . 4 (𝐴 ∈ Fin β†’ 𝒫 𝐴 ∈ Fin)
41, 3elind 4194 . . 3 (𝐴 ∈ Fin β†’ 𝒫 𝐴 ∈ (Top ∩ Fin))
5 fincmp 22896 . . 3 (𝒫 𝐴 ∈ (Top ∩ Fin) β†’ 𝒫 𝐴 ∈ Comp)
64, 5syl 17 . 2 (𝐴 ∈ Fin β†’ 𝒫 𝐴 ∈ Comp)
7 simpr 485 . . . . . . . 8 ((𝒫 𝐴 ∈ Comp ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ 𝐴)
87snssd 4812 . . . . . . 7 ((𝒫 𝐴 ∈ Comp ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯} βŠ† 𝐴)
9 vsnex 5429 . . . . . . . 8 {π‘₯} ∈ V
109elpw 4606 . . . . . . 7 ({π‘₯} ∈ 𝒫 𝐴 ↔ {π‘₯} βŠ† 𝐴)
118, 10sylibr 233 . . . . . 6 ((𝒫 𝐴 ∈ Comp ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯} ∈ 𝒫 𝐴)
1211fmpttd 7114 . . . . 5 (𝒫 𝐴 ∈ Comp β†’ (π‘₯ ∈ 𝐴 ↦ {π‘₯}):π΄βŸΆπ’« 𝐴)
1312frnd 6725 . . . 4 (𝒫 𝐴 ∈ Comp β†’ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) βŠ† 𝒫 𝐴)
14 eqid 2732 . . . . . . . 8 (π‘₯ ∈ 𝐴 ↦ {π‘₯}) = (π‘₯ ∈ 𝐴 ↦ {π‘₯})
1514rnmpt 5954 . . . . . . 7 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = {π‘₯}}
1615unieqi 4921 . . . . . 6 βˆͺ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = {π‘₯}}
179dfiun2 5036 . . . . . 6 βˆͺ π‘₯ ∈ 𝐴 {π‘₯} = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = {π‘₯}}
18 iunid 5063 . . . . . 6 βˆͺ π‘₯ ∈ 𝐴 {π‘₯} = 𝐴
1916, 17, 183eqtr2ri 2767 . . . . 5 𝐴 = βˆͺ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯})
2019a1i 11 . . . 4 (𝒫 𝐴 ∈ Comp β†’ 𝐴 = βˆͺ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}))
21 unipw 5450 . . . . . 6 βˆͺ 𝒫 𝐴 = 𝐴
2221eqcomi 2741 . . . . 5 𝐴 = βˆͺ 𝒫 𝐴
2322cmpcov 22892 . . . 4 ((𝒫 𝐴 ∈ Comp ∧ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) βŠ† 𝒫 𝐴 ∧ 𝐴 = βˆͺ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯})) β†’ βˆƒπ‘¦ ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin)𝐴 = βˆͺ 𝑦)
2413, 20, 23mpd3an23 1463 . . 3 (𝒫 𝐴 ∈ Comp β†’ βˆƒπ‘¦ ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin)𝐴 = βˆͺ 𝑦)
25 elinel2 4196 . . . . . 6 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ 𝑦 ∈ Fin)
26 elinel1 4195 . . . . . . . 8 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ 𝑦 ∈ 𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}))
2726elpwid 4611 . . . . . . 7 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ 𝑦 βŠ† ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}))
28 snfi 9043 . . . . . . . . . 10 {π‘₯} ∈ Fin
2928rgenw 3065 . . . . . . . . 9 βˆ€π‘₯ ∈ 𝐴 {π‘₯} ∈ Fin
3014fmpt 7109 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝐴 {π‘₯} ∈ Fin ↔ (π‘₯ ∈ 𝐴 ↦ {π‘₯}):𝐴⟢Fin)
3129, 30mpbi 229 . . . . . . . 8 (π‘₯ ∈ 𝐴 ↦ {π‘₯}):𝐴⟢Fin
32 frn 6724 . . . . . . . 8 ((π‘₯ ∈ 𝐴 ↦ {π‘₯}):𝐴⟢Fin β†’ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) βŠ† Fin)
3331, 32mp1i 13 . . . . . . 7 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) βŠ† Fin)
3427, 33sstrd 3992 . . . . . 6 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ 𝑦 βŠ† Fin)
35 unifi 9340 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝑦 βŠ† Fin) β†’ βˆͺ 𝑦 ∈ Fin)
3625, 34, 35syl2anc 584 . . . . 5 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ βˆͺ 𝑦 ∈ Fin)
37 eleq1 2821 . . . . 5 (𝐴 = βˆͺ 𝑦 β†’ (𝐴 ∈ Fin ↔ βˆͺ 𝑦 ∈ Fin))
3836, 37syl5ibrcom 246 . . . 4 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ (𝐴 = βˆͺ 𝑦 β†’ 𝐴 ∈ Fin))
3938rexlimiv 3148 . . 3 (βˆƒπ‘¦ ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin)𝐴 = βˆͺ 𝑦 β†’ 𝐴 ∈ Fin)
4024, 39syl 17 . 2 (𝒫 𝐴 ∈ Comp β†’ 𝐴 ∈ Fin)
416, 40impbii 208 1 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  βˆͺ ciun 4997   ↦ cmpt 5231  ran crn 5677  βŸΆwf 6539  Fincfn 8938  Topctop 22394  Compccmp 22889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-en 8939  df-fin 8942  df-top 22395  df-cmp 22890
This theorem is referenced by:  disllycmp  23001  xkohaus  23156  xkoptsub  23157  xkopt  23158
  Copyright terms: Public domain W3C validator