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Theorem discmp 22772
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 22368 . . . 4 (𝐴 ∈ Fin β†’ 𝒫 𝐴 ∈ Top)
2 pwfi 9128 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32biimpi 215 . . . 4 (𝐴 ∈ Fin β†’ 𝒫 𝐴 ∈ Fin)
41, 3elind 4158 . . 3 (𝐴 ∈ Fin β†’ 𝒫 𝐴 ∈ (Top ∩ Fin))
5 fincmp 22767 . . 3 (𝒫 𝐴 ∈ (Top ∩ Fin) β†’ 𝒫 𝐴 ∈ Comp)
64, 5syl 17 . 2 (𝐴 ∈ Fin β†’ 𝒫 𝐴 ∈ Comp)
7 simpr 486 . . . . . . . 8 ((𝒫 𝐴 ∈ Comp ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ 𝐴)
87snssd 4773 . . . . . . 7 ((𝒫 𝐴 ∈ Comp ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯} βŠ† 𝐴)
9 vsnex 5390 . . . . . . . 8 {π‘₯} ∈ V
109elpw 4568 . . . . . . 7 ({π‘₯} ∈ 𝒫 𝐴 ↔ {π‘₯} βŠ† 𝐴)
118, 10sylibr 233 . . . . . 6 ((𝒫 𝐴 ∈ Comp ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯} ∈ 𝒫 𝐴)
1211fmpttd 7067 . . . . 5 (𝒫 𝐴 ∈ Comp β†’ (π‘₯ ∈ 𝐴 ↦ {π‘₯}):π΄βŸΆπ’« 𝐴)
1312frnd 6680 . . . 4 (𝒫 𝐴 ∈ Comp β†’ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) βŠ† 𝒫 𝐴)
14 eqid 2733 . . . . . . . 8 (π‘₯ ∈ 𝐴 ↦ {π‘₯}) = (π‘₯ ∈ 𝐴 ↦ {π‘₯})
1514rnmpt 5914 . . . . . . 7 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = {π‘₯}}
1615unieqi 4882 . . . . . 6 βˆͺ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = {π‘₯}}
179dfiun2 4997 . . . . . 6 βˆͺ π‘₯ ∈ 𝐴 {π‘₯} = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = {π‘₯}}
18 iunid 5024 . . . . . 6 βˆͺ π‘₯ ∈ 𝐴 {π‘₯} = 𝐴
1916, 17, 183eqtr2ri 2768 . . . . 5 𝐴 = βˆͺ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯})
2019a1i 11 . . . 4 (𝒫 𝐴 ∈ Comp β†’ 𝐴 = βˆͺ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}))
21 unipw 5411 . . . . . 6 βˆͺ 𝒫 𝐴 = 𝐴
2221eqcomi 2742 . . . . 5 𝐴 = βˆͺ 𝒫 𝐴
2322cmpcov 22763 . . . 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 1464 . . 3 (𝒫 𝐴 ∈ Comp β†’ βˆƒπ‘¦ ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin)𝐴 = βˆͺ 𝑦)
25 elinel2 4160 . . . . . 6 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ 𝑦 ∈ Fin)
26 elinel1 4159 . . . . . . . 8 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ 𝑦 ∈ 𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}))
2726elpwid 4573 . . . . . . 7 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ 𝑦 βŠ† ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}))
28 snfi 8994 . . . . . . . . . 10 {π‘₯} ∈ Fin
2928rgenw 3065 . . . . . . . . 9 βˆ€π‘₯ ∈ 𝐴 {π‘₯} ∈ Fin
3014fmpt 7062 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝐴 {π‘₯} ∈ Fin ↔ (π‘₯ ∈ 𝐴 ↦ {π‘₯}):𝐴⟢Fin)
3129, 30mpbi 229 . . . . . . . 8 (π‘₯ ∈ 𝐴 ↦ {π‘₯}):𝐴⟢Fin
32 frn 6679 . . . . . . . 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 3958 . . . . . 6 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ 𝑦 βŠ† Fin)
35 unifi 9291 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝑦 βŠ† Fin) β†’ βˆͺ 𝑦 ∈ Fin)
3625, 34, 35syl2anc 585 . . . . 5 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ βˆͺ 𝑦 ∈ Fin)
37 eleq1 2822 . . . . 5 (𝐴 = βˆͺ 𝑦 β†’ (𝐴 ∈ Fin ↔ βˆͺ 𝑦 ∈ Fin))
3836, 37syl5ibrcom 247 . . . 4 (𝑦 ∈ (𝒫 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) ∩ Fin) β†’ (𝐴 = βˆͺ 𝑦 β†’ 𝐴 ∈ Fin))
3938rexlimiv 3142 . . 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869  βˆͺ ciun 4958   ↦ cmpt 5192  ran crn 5638  βŸΆwf 6496  Fincfn 8889  Topctop 22265  Compccmp 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7807  df-1o 8416  df-en 8890  df-fin 8893  df-top 22266  df-cmp 22761
This theorem is referenced by:  disllycmp  22872  xkohaus  23027  xkoptsub  23028  xkopt  23029
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