Theorem discmp 22457

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 22053	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2		pwfi 8923	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
3	2	biimpi 215	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
4	1, 3	elind 4124	. . 3 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5		fincmp 22452	. . 3 ⊢ (𝒫 𝐴 ∈ (Top ∩ Fin) → 𝒫 𝐴 ∈ Comp)
6	4, 5	syl 17	. 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Comp)
7		simpr 484	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	7	snssd 4739	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴)
9		snex 5349	. . . . . . . 8 ⊢ {𝑥} ∈ V
10	9	elpw 4534	. . . . . . 7 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
11	8, 10	sylibr 233	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ 𝒫 𝐴)
12	11	fmpttd 6971	. . . . 5 ⊢ (𝒫 𝐴 ∈ Comp → (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
13	12	frnd 6592	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14		eqid 2738	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
15	14	rnmpt 5853	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
16	15	unieqi 4849	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
17	9	dfiun2 4959	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
18		iunid 4986	. . . . . 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 5360	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
22	21	eqcomi 2747	. . . . 5 ⊢ 𝐴 = ∪ 𝒫 𝐴
23	22	cmpcov 22448	. . . 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 1461	. . 3 ⊢ (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = ∪ 𝑦)
25		elinel2 4126	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26		elinel1 4125	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
27	26	elpwid 4541	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
28		snfi 8788	. . . . . . . . . 10 ⊢ {𝑥} ∈ Fin
29	28	rgenw 3075	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin
30	14	fmpt 6966	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin ↔ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin)
31	29, 30	mpbi 229	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin
32		frn 6591	. . . . . . . 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 3927	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35		unifi 9038	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ⊆ Fin) → ∪ 𝑦 ∈ Fin)
36	25, 34, 35	syl2anc 583	. . . . 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 3208	. . 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 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836  ∪ ciun 4921   ↦ cmpt 5153  ran crn 5581  ⟶wf 6414  Fincfn 8691  Topctop 21950  Compccmp 22445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-en 8692  df-fin 8695  df-top 21951  df-cmp 22446

This theorem is referenced by:  disllycmp  22557  xkohaus  22712  xkoptsub  22713  xkopt  22714