Theorem islocfin 22576

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 22566	. . . 4 ⊢ LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ (∪ 𝑗 = ∪ 𝑦 ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
2	1	mptrcl 6866	. . 3 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
3		eqimss2 3974	. . . . . . . . . . 11 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
4		sspwuni 5025	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
5	3, 4	sylibr 233	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
6		velpw 4535	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 ↔ 𝑦 ⊆ 𝒫 𝑋)
7	5, 6	sylibr 233	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ∈ 𝒫 𝒫 𝑋)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) → 𝑦 ∈ 𝒫 𝒫 𝑋)
9	8	abssi 3999	. . . . . . 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 21963	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
12		pwexg 5296	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
13		pwexg 5296	. . . . . . . 8 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
14	11, 12, 13	3syl 18	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
15		ssexg 5242	. . . . . . 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 586	. . . . . 6 ⊢ (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ∈ V)
17		unieq 4847	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
18	17, 10	eqtr4di 2797	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
19	18	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
20		rexeq 3334	. . . . . . . . . 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 3327	. . . . . . . . 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 630	. . . . . . . 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 2808	. . . . . . 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 6855	. . . . . 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 683	. . . . 5 ⊢ (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
26	25	eleq2d 2824	. . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}))
27		elex 3440	. . . . . 6 ⊢ (𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} → 𝐴 ∈ V)
28	27	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}) → 𝐴 ∈ V)
29		simpr 484	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
30		islocfin.2	. . . . . . . . . 10 ⊢ 𝑌 = ∪ 𝐴
31	29, 30	eqtrdi 2795	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = ∪ 𝐴)
32	11	adantr 480	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐽)
33	31, 32	eqeltrrd 2840	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ 𝐽)
34	33	elexd 3442	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V)
35		uniexb 7592	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
36	34, 35	sylibr 233	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
37	36	adantrr 713	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
38		unieq 4847	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
39	38, 30	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = 𝑌)
40	39	eqeq2d 2749	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = 𝑌))
41		rabeq 3408	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
42	41	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ({𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
43	42	anbi2d 628	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
44	43	rexbidv 3225	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
45	44	ralbidv 3120	. . . . . . 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 630	. . . . . 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 3600	. . . . 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 798	. . . 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 278	. . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
50	2, 49	biadanii 818	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
51		3anass 1093	. 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 277	1 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ‘cfv 6418  Fincfn 8691  Topctop 21950  LocFinclocfin 22563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-top 21951  df-locfin 22566

This theorem is referenced by:  finlocfin  22579  locfintop  22580  locfinbas  22581  locfinnei  22582  lfinun  22584  dissnlocfin  22588  locfindis  22589  locfincf  22590  locfinreflem  31692  locfinref  31693