Theorem locfindis 23025

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 23019	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2		unipw 5449	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
3	2	eqcomi 2741	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
4		locfindis.1	. . . 4 ⊢ 𝑌 = ∪ 𝐶
5	3, 4	locfinbas 23017	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝑋 = 𝑌)
6	1, 5	jca 512	. 2 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
7		simpr 485	. . . . 5 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
8		uniexg 7726	. . . . . . 7 ⊢ (𝐶 ∈ PtFin → ∪ 𝐶 ∈ V)
9	4, 8	eqeltrid 2837	. . . . . 6 ⊢ (𝐶 ∈ PtFin → 𝑌 ∈ V)
10	9	adantr 481	. . . . 5 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
11	7, 10	eqeltrd 2833	. . . 4 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12		distop 22489	. . . 4 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
13	11, 12	syl 17	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14		snelpwi 5442	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
15	14	adantl 482	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
16		snidg 4661	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ {𝑥})
17	16	adantl 482	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {𝑥})
18		simpll 765	. . . . . 6 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ PtFin)
19	7	eleq2d 2819	. . . . . . 7 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ 𝑌))
20	19	biimpa 477	. . . . . 6 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑌)
21	4	ptfinfin 23014	. . . . . 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 2822	. . . . . . 7 ⊢ (𝑦 = {𝑥} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑥}))
24		ineq2 4205	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑥} → (𝑠 ∩ 𝑦) = (𝑠 ∩ {𝑥}))
25	24	neeq1d 3000	. . . . . . . . . 10 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26		disjsn 4714	. . . . . . . . . . 11 ⊢ ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑠)
27	26	necon2abii 2991	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
28	25, 27	bitr4di 288	. . . . . . . . 9 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ 𝑥 ∈ 𝑠))
29	28	rabbidv 3440	. . . . . . . 8 ⊢ (𝑦 = {𝑥} → {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} = {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠})
30	29	eleq1d 2818	. . . . . . 7 ⊢ (𝑦 = {𝑥} → ({𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin))
31	23, 30	anbi12d 631	. . . . . 6 ⊢ (𝑦 = {𝑥} → ((𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)))
32	31	rspcev 3612	. . . . 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 835	. . . 4 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
34	33	ralrimiva 3146	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
35	3, 4	islocfin 23012	. . 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 208	1 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ∩ cin 3946  ∅c0 4321  𝒫 cpw 4601  {csn 4627  ∪ cuni 4907  ‘cfv 6540  Fincfn 8935  Topctop 22386  PtFincptfin 22998  LocFinclocfin 22999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-en 8936  df-fin 8939  df-top 22387  df-ptfin 23001  df-locfin 23002

This theorem is referenced by: (None)