Theorem islocfin 23642

Description: The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypotheses

Ref	Expression
islocfin.1	⊢ 𝑋 = ∪ 𝐽
islocfin.2	⊢ 𝑌 = ∪ 𝐴

Assertion

Ref	Expression
islocfin	⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))

Distinct variable groups:   𝑛,𝑠,𝑥,𝐴   𝑛,𝐽,𝑥   𝑥,𝑋

Allowed substitution hints:   𝐽(𝑠)   𝑋(𝑛,𝑠)   𝑌(𝑥,𝑛,𝑠)

Proof of Theorem islocfin

Dummy variables 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-locfin 23632	. . . 4 ⊢ LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ (∪ 𝑗 = ∪ 𝑦 ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
2	1	mptrcl 7000	. . 3 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
3		eqimss2 4004	. . . . . . . . . . 11 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
4		sspwuni 5070	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
5	3, 4	sylibr 237	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
6		velpw 4572	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 ↔ 𝑦 ⊆ 𝒫 𝑋)
7	5, 6	sylibr 237	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ∈ 𝒫 𝒫 𝑋)
8	7	adantr 485	. . . . . . . 8 ⊢ ((𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) → 𝑦 ∈ 𝒫 𝒫 𝑋)
9	8	abssi 4030	. . . . . . 7 ⊢ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ⊆ 𝒫 𝒫 𝑋
10		islocfin.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
11	10	topopn 23031	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
12		pwexg 5350	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
13		pwexg 5350	. . . . . . . 8 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
14	11, 12, 13	3syl 19	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
15		ssexg 5294	. . . . . . 7 ⊢ (({𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ⊆ 𝒫 𝒫 𝑋 ∧ 𝒫 𝒫 𝑋 ∈ V) → {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ∈ V)
16	9, 14, 15	sylancr 598	. . . . . 6 ⊢ (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ∈ V)
17		unieq 4887	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
18	17, 10	eqtr4di 2822	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
19	18	eqeq1d 2771	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
20		rexeq 3325	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
21	18, 20	raleqbidv 3345	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ ∪ 𝑗∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
22	19, 21	anbi12d 643	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 = ∪ 𝑦 ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) ↔ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
23	22	abbidv 2835	. . . . . . 7 ⊢ (𝑗 = 𝐽 → {𝑦 ∣ (∪ 𝑗 = ∪ 𝑦 ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} = {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
24	23, 1	fvmptg 6988	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ∈ V) → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
25	16, 24	mpdan 699	. . . . 5 ⊢ (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
26	25	eleq2d 2855	. . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}))
27		elex 3484	. . . . . 6 ⊢ (𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} → 𝐴 ∈ V)
28	27	adantl 486	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}) → 𝐴 ∈ V)
29		simpr 489	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
30		islocfin.2	. . . . . . . . . 10 ⊢ 𝑌 = ∪ 𝐴
31	29, 30	eqtrdi 2820	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = ∪ 𝐴)
32	11	adantr 485	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐽)
33	31, 32	eqeltrrd 2870	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ 𝐽)
34	33	elexd 3486	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V)
35		uniexb 7762	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
36	34, 35	sylibr 237	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
37	36	adantrr 729	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
38		unieq 4887	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
39	38, 30	eqtr4di 2822	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = 𝑌)
40	39	eqeq2d 2780	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = 𝑌))
41		rabeq 3437	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
42	41	eleq1d 2854	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ({𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
43	42	anbi2d 641	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
44	43	rexbidv 3195	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
45	44	ralbidv 3194	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
46	40, 45	anbi12d 643	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
47	46	elabg 3644	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
48	28, 37, 47	pm5.21nd 813	. . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
49	26, 48	bitrd 282	. . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
50	2, 49	biadanii 833	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
51		3anass 1109	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
52	50, 51	bitr4i 281	1 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876  ‘cfv 6537  Fincfn 8942  Topctop 23018  LocFinclocfin 23629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-top 23019  df-locfin 23632

This theorem is referenced by:  finlocfin  23645  locfintop  23646  locfinbas  23647  locfinnei  23648  lfinun  23650  dissnlocfin  23654  locfindis  23655  locfincf  23656  locfinreflem  34174  locfinref  34175