Theorem locfindis 23483

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 23477	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2		unipw 5391	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
3	2	eqcomi 2744	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
4		locfindis.1	. . . 4 ⊢ 𝑌 = ∪ 𝐶
5	3, 4	locfinbas 23475	. . 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 7683	. . . . . . 7 ⊢ (𝐶 ∈ PtFin → ∪ 𝐶 ∈ V)
9	4, 8	eqeltrid 2839	. . . . . 6 ⊢ (𝐶 ∈ PtFin → 𝑌 ∈ V)
10	9	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
11	7, 10	eqeltrd 2835	. . . 4 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12		distop 22948	. . . 4 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
13	11, 12	syl 17	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14		snelpwi 5385	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
15	14	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
16		snidg 4594	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ {𝑥})
17	16	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {𝑥})
18		simpll 767	. . . . . 6 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ PtFin)
19	7	eleq2d 2821	. . . . . . 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 23472	. . . . . 6 ⊢ ((𝐶 ∈ PtFin ∧ 𝑥 ∈ 𝑌) → {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)
22	18, 20, 21	syl2anc 585	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)
23		eleq2 2824	. . . . . . 7 ⊢ (𝑦 = {𝑥} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑥}))
24		ineq2 4145	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑥} → (𝑠 ∩ 𝑦) = (𝑠 ∩ {𝑥}))
25	24	neeq1d 2989	. . . . . . . . . 10 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26		disjsn 4645	. . . . . . . . . . 11 ⊢ ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑠)
27	26	necon2abii 2980	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
28	25, 27	bitr4di 289	. . . . . . . . 9 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ 𝑥 ∈ 𝑠))
29	28	rabbidv 3394	. . . . . . . 8 ⊢ (𝑦 = {𝑥} → {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} = {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠})
30	29	eleq1d 2820	. . . . . . 7 ⊢ (𝑦 = {𝑥} → ({𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin))
31	23, 30	anbi12d 633	. . . . . 6 ⊢ (𝑦 = {𝑥} → ((𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)))
32	31	rspcev 3562	. . . . 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 837	. . . 4 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
34	33	ralrimiva 3127	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
35	3, 4	islocfin 23470	. . 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 1345	. 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 1542   ∈ wcel 2114   ≠ wne 2930  ∀wral 3049  ∃wrex 3059  {crab 3387  Vcvv 3427   ∩ cin 3884  ∅c0 4263  𝒫 cpw 4531  {csn 4557  ∪ cuni 4840  ‘cfv 6487  Fincfn 8882  Topctop 22846  PtFincptfin 23456  LocFinclocfin 23457

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-om 7807  df-1o 8394  df-en 8883  df-fin 8886  df-top 22847  df-ptfin 23459  df-locfin 23460

This theorem is referenced by: (None)