Theorem discmp 23438

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 23035	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Top)
2		pwfi 9259	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
3	2	biimpi 218	. . . 4 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
4	1, 3	elind 4152	. . 3 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ (Top ∩ Fin))
5		fincmp 23433	. . 3 ⊢ (𝒫 𝐴 ∈ (Top ∩ Fin) → 𝒫 𝐴 ∈ Comp)
6	4, 5	syl 17	. 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Comp)
7		simpr 488	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	7	snssd 4744	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴)
9		vsnex 5391	. . . . . . . 8 ⊢ {𝑥} ∈ V
10	9	elpw 4558	. . . . . . 7 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
11	8, 10	sylibr 236	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ Comp ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ 𝒫 𝐴)
12	11	fmpttd 7092	. . . . 5 ⊢ (𝒫 𝐴 ∈ Comp → (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶𝒫 𝐴)
13	12	frnd 6696	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ⊆ 𝒫 𝐴)
14		eqid 2761	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
15	14	rnmpt 5931	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
16	15	unieqi 4876	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
17	9	dfiun2 4988	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
18		iunid 5017	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
19	16, 17, 18	3eqtr2ri 2791	. . . . 5 ⊢ 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥})
20	19	a1i 11	. . . 4 ⊢ (𝒫 𝐴 ∈ Comp → 𝐴 = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
21		unipw 5416	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
22	21	eqcomi 2770	. . . . 5 ⊢ 𝐴 = ∪ 𝒫 𝐴
23	22	cmpcov 23429	. . . 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 1483	. . 3 ⊢ (𝒫 𝐴 ∈ Comp → ∃𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin)𝐴 = ∪ 𝑦)
25		elinel2 4154	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ Fin)
26		elinel1 4153	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ∈ 𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
27	26	elpwid 4563	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ ran (𝑥 ∈ 𝐴 ↦ {𝑥}))
28		snfi 9020	. . . . . . . . . 10 ⊢ {𝑥} ∈ Fin
29	28	rgenw 3079	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin
30	14	fmpt 7087	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 {𝑥} ∈ Fin ↔ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin)
31	29, 30	mpbi 232	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}):𝐴⟶Fin
32		frn 6695	. . . . . . . 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 3946	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → 𝑦 ⊆ Fin)
35		unifi 9284	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ⊆ Fin) → ∪ 𝑦 ∈ Fin)
36	25, 34, 35	syl2anc 593	. . . . 5 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → ∪ 𝑦 ∈ Fin)
37		eleq1 2849	. . . . 5 ⊢ (𝐴 = ∪ 𝑦 → (𝐴 ∈ Fin ↔ ∪ 𝑦 ∈ Fin))
38	36, 37	syl5ibrcom 249	. . . 4 ⊢ (𝑦 ∈ (𝒫 ran (𝑥 ∈ 𝐴 ↦ {𝑥}) ∩ Fin) → (𝐴 = ∪ 𝑦 → 𝐴 ∈ Fin))
39	38	rexlimiv 3155	. . 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 211	1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4554  {csn 4581  ∪ cuni 4864  ∪ ciun 4948   ↦ cmpt 5180  ran crn 5646  ⟶wf 6513  Fincfn 8923  Topctop 22933  Compccmp 23426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-om 7843  df-1o 8432  df-en 8924  df-dom 8925  df-fin 8927  df-top 22934  df-cmp 23427

This theorem is referenced by:  disllycmp  23538  xkohaus  23693  xkoptsub  23694  xkopt  23695