Theorem discmp 22549

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.

Step	Hyp	Ref	Expression
1		distop 22145	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2		pwfi 8961	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
3	2	biimpi 215	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
4	1, 3	elind 4128	. . 3 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5		fincmp 22544	. . 3 ⊢ (𝒫 𝐴 ∈ (Top ∩ Fin) → 𝒫 𝐴 ∈ Comp)
6	4, 5	syl 17	. 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Comp)
7		simpr 485	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	7	snssd 4742	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴)
9		snex 5354	. . . . . . . 8 ⊢ {𝑥} ∈ V
10	9	elpw 4537	. . . . . . 7 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
11	8, 10	sylibr 233	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ 𝒫 𝐴)
12	11	fmpttd 6989	. . . . 5 ⊢ (𝒫 𝐴 ∈ Comp → (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
13	12	frnd 6608	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14		eqid 2738	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
15	14	rnmpt 5864	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
16	15	unieqi 4852	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
17	9	dfiun2 4963	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
18		iunid 4990	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
19	16, 17, 18	3eqtr2ri 2773	. . . . 5 ⊢ 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥})
20	19	a1i 11	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
21		unipw 5366	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
22	21	eqcomi 2747	. . . . 5 ⊢ 𝐴 = ∪ 𝒫 𝐴
23	22	cmpcov 22540	. . . 4 ⊢ ((𝒫 𝐴 ∈ Comp ∧ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴 ∧ 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥})) → ∃𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = ∪ 𝑦)
24	13, 20, 23	mpd3an23 1462	. . 3 ⊢ (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = ∪ 𝑦)
25		elinel2 4130	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26		elinel1 4129	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
27	26	elpwid 4544	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
28		snfi 8834	. . . . . . . . . 10 ⊢ {𝑥} ∈ Fin
29	28	rgenw 3076	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin
30	14	fmpt 6984	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin ↔ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin)
31	29, 30	mpbi 229	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin
32		frn 6607	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin → ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ⊆ Fin)
33	31, 32	mp1i 13	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ⊆ Fin)
34	27, 33	sstrd 3931	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35		unifi 9108	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ⊆ Fin) → ∪ 𝑦 ∈ Fin)
36	25, 34, 35	syl2anc 584	. . . . 5 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → ∪ 𝑦 ∈ Fin)
37		eleq1 2826	. . . . 5 ⊢ (𝐴 = ∪ 𝑦 → (𝐴 ∈ Fin ↔ ∪ 𝑦 ∈ Fin))
38	36, 37	syl5ibrcom 246	. . . 4 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → (𝐴 = ∪ 𝑦 → 𝐴 ∈ Fin))
39	38	rexlimiv 3209	. . 3 ⊢ (∃𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = ∪ 𝑦 → 𝐴 ∈ Fin)
40	24, 39	syl 17	. 2 ⊢ (𝒫 𝐴 ∈ Comp → 𝐴 ∈ Fin)
41	6, 40	impbii 208	1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839  ∪ ciun 4924   ↦ cmpt 5157  ran crn 5590  ⟶wf 6429  Fincfn 8733  Topctop 22042  Compccmp 22537

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-en 8734  df-fin 8737  df-top 22043  df-cmp 22538

This theorem is referenced by:  disllycmp  22649  xkohaus  22804  xkoptsub  22805  xkopt  22806