Theorem discmp 23314

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 22911	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2		pwfi 9210	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
3	2	biimpi 216	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
4	1, 3	elind 4149	. . 3 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5		fincmp 23309	. . 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 4760	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴)
9		vsnex 5374	. . . . . . . 8 ⊢ {𝑥} ∈ V
10	9	elpw 4553	. . . . . . 7 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
11	8, 10	sylibr 234	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ 𝒫 𝐴)
12	11	fmpttd 7054	. . . . 5 ⊢ (𝒫 𝐴 ∈ Comp → (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
13	12	frnd 6664	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14		eqid 2733	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
15	14	rnmpt 5901	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
16	15	unieqi 4870	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
17	9	dfiun2 4982	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
18		iunid 5011	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
19	16, 17, 18	3eqtr2ri 2763	. . . . 5 ⊢ 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥})
20	19	a1i 11	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
21		unipw 5393	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
22	21	eqcomi 2742	. . . . 5 ⊢ 𝐴 = ∪ 𝒫 𝐴
23	22	cmpcov 23305	. . . 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 4151	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26		elinel1 4150	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
27	26	elpwid 4558	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
28		snfi 8972	. . . . . . . . . 10 ⊢ {𝑥} ∈ Fin
29	28	rgenw 3052	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin
30	14	fmpt 7049	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin ↔ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin)
31	29, 30	mpbi 230	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin
32		frn 6663	. . . . . . . 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 3941	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35		unifi 9235	. . . . . 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 2821	. . . . 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 3127	. . 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 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  ∃wrex 3057   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4549  {csn 4575  ∪ cuni 4858  ∪ ciun 4941   ↦ cmpt 5174  ran crn 5620  ⟶wf 6482  Fincfn 8875  Topctop 22809  Compccmp 23302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7803  df-1o 8391  df-en 8876  df-dom 8877  df-fin 8879  df-top 22810  df-cmp 23303

This theorem is referenced by:  disllycmp  23414  xkohaus  23569  xkoptsub  23570  xkopt  23571