Theorem locfindis 23796

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 23790	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2		unipw 5418	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
3	2	eqcomi 2769	. . . 4 ⊢ 𝑋 = ∪ 𝒫 𝑋
4		locfindis.1	. . . 4 ⊢ 𝑌 = ∪ 𝐶
5	3, 4	locfinbas 23788	. . 3 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝑋 = 𝑌)
6	1, 5	jca 521	. 2 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) → (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
7		simpr 490	. . . . 5 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
8		uniexg 7741	. . . . . . 7 ⊢ (𝐶 ∈ PtFin → ∪ 𝐶 ∈ V)
9	4, 8	eqeltrid 2864	. . . . . 6 ⊢ (𝐶 ∈ PtFin → 𝑌 ∈ V)
10	9	adantr 486	. . . . 5 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
11	7, 10	eqeltrd 2860	. . . 4 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12		distop 23260	. . . 4 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
13	11, 12	syl 18	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14		snelpwi 5412	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
15	14	adantl 487	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ 𝒫 𝑋)
16		snidg 4621	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ {𝑥})
17	16	adantl 487	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {𝑥})
18		simpll 779	. . . . . 6 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ PtFin)
19	7	eleq2d 2846	. . . . . . 7 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ 𝑌))
20	19	biimpa 482	. . . . . 6 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑌)
21	4	ptfinfin 23785	. . . . . 6 ⊢ ((𝐶 ∈ PtFin ∧ 𝑥 ∈ 𝑌) → {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)
22	18, 20, 21	syl2anc 596	. . . . 5 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)
23		eleq2 2849	. . . . . . 7 ⊢ (𝑦 = {𝑥} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑥}))
24		ineq2 4160	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑥} → (𝑠 ∩ 𝑦) = (𝑠 ∩ {𝑥}))
25	24	neeq1d 3014	. . . . . . . . . 10 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26		disjsn 4672	. . . . . . . . . . 11 ⊢ ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑠)
27	26	necon2abii 3005	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
28	25, 27	bitr4di 292	. . . . . . . . 9 ⊢ (𝑦 = {𝑥} → ((𝑠 ∩ 𝑦) ≠ ∅ ↔ 𝑥 ∈ 𝑠))
29	28	rabbidv 3419	. . . . . . . 8 ⊢ (𝑦 = {𝑥} → {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} = {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠})
30	29	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = {𝑥} → ({𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin))
31	23, 30	anbi12d 644	. . . . . 6 ⊢ (𝑦 = {𝑥} → ((𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ∈ 𝑠} ∈ Fin)))
32	31	rspcev 3576	. . . . 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 850	. . . 4 ⊢ (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
34	33	ralrimiva 3154	. . 3 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝑋(𝑥 ∈ 𝑦 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑦) ≠ ∅} ∈ Fin))
35	3, 4	islocfin 23783	. . 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 1362	. 2 ⊢ ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝒫 𝑋))
37	6, 36	impbii 212	1 ⊢ (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ∩ cin 3898  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867  ‘cfv 6528  Fincfn 8952  Topctop 23158  PtFincptfin 23769  LocFinclocfin 23770

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-om 7862  df-1o 8455  df-en 8953  df-fin 8956  df-top 23159  df-ptfin 23772  df-locfin 23773

This theorem is used by: (None)