Theorem islocfin 23891

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 23881	. . . 4 ⊢ LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ (∪ 𝑗 = ∪ 𝑦 ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
2	1	mptrcl 7034	. . 3 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
3		eqimss2 3972	. . . . . . . . . . 11 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
4		sspwuni 5065	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
5	3, 4	sylibr 237	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
6		velpw 4565	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 ↔ 𝑦 ⊆ 𝒫 𝑋)
7	5, 6	sylibr 237	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ∈ 𝒫 𝒫 𝑋)
8	7	adantr 486	. . . . . . . 8 ⊢ ((𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) → 𝑦 ∈ 𝒫 𝒫 𝑋)
9	8	abssi 3996	. . . . . . 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 23279	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
12		pwexg 5347	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
13		pwexg 5347	. . . . . . . 8 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
14	11, 12, 13	3syl 19	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
15		ssexg 5288	. . . . . . 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 599	. . . . . 6 ⊢ (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ∈ V)
17		unieq 4883	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
18	17, 10	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
19	18	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
20		rexeq 3319	. . . . . . . . . 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 3338	. . . . . . . . 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 644	. . . . . . . 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 2829	. . . . . . 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 7022	. . . . . 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 700	. . . . 5 ⊢ (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
26	25	eleq2d 2850	. . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}))
27		elex 3475	. . . . . 6 ⊢ (𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} → 𝐴 ∈ V)
28	27	adantl 487	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}) → 𝐴 ∈ V)
29		simpr 490	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
30		islocfin.2	. . . . . . . . . 10 ⊢ 𝑌 = ∪ 𝐴
31	29, 30	eqtrdi 2814	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = ∪ 𝐴)
32	11	adantr 486	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐽)
33	31, 32	eqeltrrd 2865	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ 𝐽)
34	33	elexd 3477	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V)
35		uniexb 7810	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
36	34, 35	sylibr 237	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
37	36	adantrr 730	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
38		unieq 4883	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
39	38, 30	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = 𝑌)
40	39	eqeq2d 2774	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = 𝑌))
41		rabeq 3430	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
42	41	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ({𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
43	42	anbi2d 642	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
44	43	rexbidv 3190	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
45	44	ralbidv 3189	. . . . . . 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 644	. . . . . 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 3633	. . . . 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 814	. . . 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 834	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
51		3anass 1111	. 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)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4872  ‘cfv 6566  Fincfn 9011  Topctop 23266  LocFinclocfin 23878

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-top 23267  df-locfin 23881

This theorem is used by:  finlocfin  23894  locfintop  23895  locfinbas  23896  locfinnei  23897  lfinun  23899  dissnlocfin  23903  locfindis  23904  locfincf  23905  locfinreflem  34583  locfinref  34584