Theorem locfindis 23539

Description: The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
locfindis.1	⊢ 𝑌 = ∪ 𝐶

Assertion

Ref	Expression
locfindis	⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))

Proof of Theorem locfindis

Dummy variables 𝑥 𝑠 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lfinpfin 23533	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2		unipw 5454	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
3	2	eqcomi 2745	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
4		locfindis.1	. . . 4 ⊢ 𝑌 = ∪ 𝐶
5	3, 4	locfinbas 23531	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝑋 = 𝑌)
6	1, 5	jca 511	. 2 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
7		simpr 484	. . . . 5 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
8		uniexg 7761	. . . . . . 7 ⊢ (𝐶 ∈ PtFin → ∪ 𝐶 ∈ V)
9	4, 8	eqeltrid 2844	. . . . . 6 ⊢ (𝐶 ∈ PtFin → 𝑌 ∈ V)
10	9	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
11	7, 10	eqeltrd 2840	. . . 4 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12		distop 23003	. . . 4 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
13	11, 12	syl 17	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14		snelpwi 5447	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
15	14	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
16		snidg 4659	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ {𝑥})
17	16	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {𝑥})
18		simpll 766	. . . . . 6 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ PtFin)
19	7	eleq2d 2826	. . . . . . 7 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ 𝑌))
20	19	biimpa 476	. . . . . 6 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑌)
21	4	ptfinfin 23528	. . . . . 6 ⊢ ((𝐶 ∈ PtFin ∧ 𝑥 ∈ 𝑌) → {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)
22	18, 20, 21	syl2anc 584	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)
23		eleq2 2829	. . . . . . 7 ⊢ (𝑦 = {𝑥} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑥}))
24		ineq2 4213	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑥} → (𝑠 ∩ 𝑦) = (𝑠 ∩ {𝑥}))
25	24	neeq1d 2999	. . . . . . . . . 10 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26		disjsn 4710	. . . . . . . . . . 11 ⊢ ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑠)
27	26	necon2abii 2990	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
28	25, 27	bitr4di 289	. . . . . . . . 9 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ 𝑥 ∈ 𝑠))
29	28	rabbidv 3443	. . . . . . . 8 ⊢ (𝑦 = {𝑥} → {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} = {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠})
30	29	eleq1d 2825	. . . . . . 7 ⊢ (𝑦 = {𝑥} → ({𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin))
31	23, 30	anbi12d 632	. . . . . 6 ⊢ (𝑦 = {𝑥} → ((𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)))
32	31	rspcev 3621	. . . . 5 ⊢ (({𝑥} ∈ 𝒫 𝑋 ∧ (𝑥 ∈ {𝑥} ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
33	15, 17, 22, 32	syl12anc 836	. . . 4 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
34	33	ralrimiva 3145	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
35	3, 4	islocfin 23526	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin)))
36	13, 7, 34, 35	syl3anbrc 1343	. 2 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝒫 𝑋))
37	6, 36	impbii 209	1 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3435  Vcvv 3479   ∩ cin 3949  ∅c0 4332  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906  ‘cfv 6560  Fincfn 8986  Topctop 22900  PtFincptfin 23512  LocFinclocfin 23513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-en 8987  df-fin 8990  df-top 22901  df-ptfin 23515  df-locfin 23516

This theorem is referenced by: (None)