Theorem locfindis 23424

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 23418	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2		unipw 5413	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
3	2	eqcomi 2739	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
4		locfindis.1	. . . 4 ⊢ 𝑌 = ∪ 𝐶
5	3, 4	locfinbas 23416	. . 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 7719	. . . . . . 7 ⊢ (𝐶 ∈ PtFin → ∪ 𝐶 ∈ V)
9	4, 8	eqeltrid 2833	. . . . . 6 ⊢ (𝐶 ∈ PtFin → 𝑌 ∈ V)
10	9	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
11	7, 10	eqeltrd 2829	. . . 4 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12		distop 22889	. . . 4 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
13	11, 12	syl 17	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14		snelpwi 5406	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
15	14	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
16		snidg 4627	. . . . . 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 2815	. . . . . . 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 23413	. . . . . 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 2818	. . . . . . 7 ⊢ (𝑦 = {𝑥} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑥}))
24		ineq2 4180	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑥} → (𝑠 ∩ 𝑦) = (𝑠 ∩ {𝑥}))
25	24	neeq1d 2985	. . . . . . . . . 10 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26		disjsn 4678	. . . . . . . . . . 11 ⊢ ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑠)
27	26	necon2abii 2976	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
28	25, 27	bitr4di 289	. . . . . . . . 9 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ 𝑥 ∈ 𝑠))
29	28	rabbidv 3416	. . . . . . . 8 ⊢ (𝑦 = {𝑥} → {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} = {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠})
30	29	eleq1d 2814	. . . . . . 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 3591	. . . . 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 3126	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
35	3, 4	islocfin 23411	. . 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 1344	. 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 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  {crab 3408  Vcvv 3450   ∩ cin 3916  ∅c0 4299  𝒫 cpw 4566  {csn 4592  ∪ cuni 4874  ‘cfv 6514  Fincfn 8921  Topctop 22787  PtFincptfin 23397  LocFinclocfin 23398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-om 7846  df-1o 8437  df-en 8922  df-fin 8925  df-top 22788  df-ptfin 23400  df-locfin 23401

This theorem is referenced by: (None)