Theorem islocfin 23495

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 23485	. . . 4 ⊢ LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ (∪ 𝑗 = ∪ 𝑦 ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
2	1	mptrcl 6952	. . 3 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
3		eqimss2 3982	. . . . . . . . . . 11 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
4		sspwuni 5043	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
5	3, 4	sylibr 234	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
6		velpw 4547	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 ↔ 𝑦 ⊆ 𝒫 𝑋)
7	5, 6	sylibr 234	. . . . . . . . 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 4009	. . . . . . 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 22884	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
12		pwexg 5316	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
13		pwexg 5316	. . . . . . . 8 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
14	11, 12, 13	3syl 18	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
15		ssexg 5261	. . . . . . 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 588	. . . . . 6 ⊢ (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ∈ V)
17		unieq 4862	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
18	17, 10	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
19	18	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
20		rexeq 3292	. . . . . . . . . 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 3312	. . . . . . . . 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 633	. . . . . . . 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 2803	. . . . . . 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 6940	. . . . . 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 688	. . . . 5 ⊢ (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
26	25	eleq2d 2823	. . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}))
27		elex 3451	. . . . . 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 2788	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = ∪ 𝐴)
32	11	adantr 480	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐽)
33	31, 32	eqeltrrd 2838	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ 𝐽)
34	33	elexd 3454	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V)
35		uniexb 7712	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
36	34, 35	sylibr 234	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
37	36	adantrr 718	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
38		unieq 4862	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
39	38, 30	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = 𝑌)
40	39	eqeq2d 2748	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = 𝑌))
41		rabeq 3404	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
42	41	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ({𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
43	42	anbi2d 631	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
44	43	rexbidv 3162	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
45	44	ralbidv 3161	. . . . . . 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 633	. . . . . 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 3620	. . . . 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 802	. . . 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 279	. . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
50	2, 49	biadanii 822	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
51		3anass 1095	. 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 278	1 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6493  Fincfn 8887  Topctop 22871  LocFinclocfin 23482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fv 6501  df-top 22872  df-locfin 23485

This theorem is referenced by:  finlocfin  23498  locfintop  23499  locfinbas  23500  locfinnei  23501  lfinun  23503  dissnlocfin  23507  locfindis  23508  locfincf  23509  locfinreflem  34003  locfinref  34004