Theorem discmp 23292

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 22889	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2		pwfi 9275	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
3	2	biimpi 216	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
4	1, 3	elind 4166	. . 3 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5		fincmp 23287	. . 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 4776	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴)
9		vsnex 5392	. . . . . . . 8 ⊢ {𝑥} ∈ V
10	9	elpw 4570	. . . . . . 7 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
11	8, 10	sylibr 234	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ 𝒫 𝐴)
12	11	fmpttd 7090	. . . . 5 ⊢ (𝒫 𝐴 ∈ Comp → (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
13	12	frnd 6699	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14		eqid 2730	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
15	14	rnmpt 5924	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
16	15	unieqi 4886	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
17	9	dfiun2 5000	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
18		iunid 5027	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
19	16, 17, 18	3eqtr2ri 2760	. . . . 5 ⊢ 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥})
20	19	a1i 11	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
21		unipw 5413	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
22	21	eqcomi 2739	. . . . 5 ⊢ 𝐴 = ∪ 𝒫 𝐴
23	22	cmpcov 23283	. . . 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 1465	. . 3 ⊢ (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = ∪ 𝑦)
25		elinel2 4168	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26		elinel1 4167	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
27	26	elpwid 4575	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
28		snfi 9017	. . . . . . . . . 10 ⊢ {𝑥} ∈ Fin
29	28	rgenw 3049	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin
30	14	fmpt 7085	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin ↔ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin)
31	29, 30	mpbi 230	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin
32		frn 6698	. . . . . . . 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 3960	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35		unifi 9302	. . . . . 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 2817	. . . . 5 ⊢ (𝐴 = ∪ 𝑦 → (𝐴 ∈ Fin ↔ ∪ 𝑦 ∈ Fin))
38	36, 37	syl5ibrcom 247	. . . 4 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → (𝐴 = ∪ 𝑦 → 𝐴 ∈ Fin))
39	38	rexlimiv 3128	. . 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 209	1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  {csn 4592  ∪ cuni 4874  ∪ ciun 4958   ↦ cmpt 5191  ran crn 5642  ⟶wf 6510  Fincfn 8921  Topctop 22787  Compccmp 23280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-om 7846  df-1o 8437  df-en 8922  df-dom 8923  df-fin 8925  df-top 22788  df-cmp 23281

This theorem is referenced by:  disllycmp  23392  xkohaus  23547  xkoptsub  23548  xkopt  23549