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Theorem discmp 21998
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 21595 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2 pwfi 8811 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32biimpi 218 . . . 4 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
41, 3elind 4169 . . 3 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5 fincmp 21993 . . 3 (𝒫 𝐴 ∈ (Top ∩ Fin) → 𝒫 𝐴 ∈ Comp)
64, 5syl 17 . 2 (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Comp)
7 simpr 487 . . . . . . . 8 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → 𝑥𝐴)
87snssd 4734 . . . . . . 7 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ⊆ 𝐴)
9 snex 5322 . . . . . . . 8 {𝑥} ∈ V
109elpw 4544 . . . . . . 7 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
118, 10sylibr 236 . . . . . 6 ((𝒫 𝐴 ∈ Comp ∧ 𝑥𝐴) → {𝑥} ∈ 𝒫 𝐴)
1211fmpttd 6872 . . . . 5 (𝒫 𝐴 ∈ Comp → (𝑥𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
1312frnd 6514 . . . 4 (𝒫 𝐴 ∈ Comp → ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14 eqid 2819 . . . . . . . 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 2849 . . . . 5 𝐴 = ran (𝑥𝐴 ↦ {𝑥})
2019a1i 11 . . . 4 (𝒫 𝐴 ∈ Comp → 𝐴 = ran (𝑥𝐴 ↦ {𝑥}))
21 unipw 5333 . . . . . 6 𝒫 𝐴 = 𝐴
2221eqcomi 2828 . . . . 5 𝐴 = 𝒫 𝐴
2322cmpcov 21989 . . . 4 ((𝒫 𝐴 ∈ Comp ∧ ran (𝑥𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴𝐴 = ran (𝑥𝐴 ↦ {𝑥})) → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
2413, 20, 23mpd3an23 1456 . . 3 (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦)
25 elinel2 4171 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26 elinel1 4170 . . . . . . . 8 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥𝐴 ↦ {𝑥}))
2726elpwid 4551 . . . . . . 7 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥𝐴 ↦ {𝑥}))
28 snfi 8586 . . . . . . . . . 10 {𝑥} ∈ Fin
2928rgenw 3148 . . . . . . . . 9 𝑥𝐴 {𝑥} ∈ Fin
3014fmpt 6867 . . . . . . . . 9 (∀𝑥𝐴 {𝑥} ∈ Fin ↔ (𝑥𝐴 ↦ {𝑥}):𝐴⟶Fin)
3129, 30mpbi 232 . . . . . . . 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 3975 . . . . . 6 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35 unifi 8805 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝑦 ⊆ Fin) → 𝑦 ∈ Fin)
3625, 34, 35syl2anc 586 . . . . 5 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
37 eleq1 2898 . . . . 5 (𝐴 = 𝑦 → (𝐴 ∈ Fin ↔ 𝑦 ∈ Fin))
3836, 37syl5ibrcom 249 . . . 4 (𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin) → (𝐴 = 𝑦𝐴 ∈ Fin))
3938rexlimiv 3278 . . 3 (∃𝑦 ∈ (𝒫 ran (𝑥𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = 𝑦𝐴 ∈ Fin)
4024, 39syl 17 . 2 (𝒫 𝐴 ∈ Comp → 𝐴 ∈ Fin)
416, 40impbii 211 1 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1530  wcel 2107  {cab 2797  wral 3136  wrex 3137  cin 3933  wss 3934  𝒫 cpw 4537  {csn 4559   cuni 4830   ciun 4910  cmpt 5137  ran crn 5549  wf 6344  Fincfn 8501  Topctop 21493  Compccmp 21986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  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 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-top 21494  df-cmp 21987
This theorem is referenced by:  disllycmp  22098  xkohaus  22253  xkoptsub  22254  xkopt  22255
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