Theorem islocfin 23380

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 23370	. . . 4 ⊢ LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ (∪ 𝑗 = ∪ 𝑦 ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑛 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
2	1	mptrcl 6959	. . 3 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
3		eqimss2 4003	. . . . . . . . . . 11 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
4		sspwuni 5059	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
5	3, 4	sylibr 234	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
6		velpw 4564	. . . . . . . . . 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 4029	. . . . . . 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 22769	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
12		pwexg 5328	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
13		pwexg 5328	. . . . . . . 8 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
14	11, 12, 13	3syl 18	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
15		ssexg 5273	. . . . . . 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 587	. . . . . 6 ⊢ (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))} ∈ V)
17		unieq 4878	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
18	17, 10	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
19	18	eqeq1d 2731	. . . . . . . . 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 3316	. . . . . . . . 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 632	. . . . . . . 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 2795	. . . . . . 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 6948	. . . . . 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 687	. . . . 5 ⊢ (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
26	25	eleq2d 2814	. . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = ∪ 𝑦 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}))
27		elex 3465	. . . . . 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 2780	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = ∪ 𝐴)
32	11	adantr 480	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐽)
33	31, 32	eqeltrrd 2829	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ 𝐽)
34	33	elexd 3468	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V)
35		uniexb 7720	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
36	34, 35	sylibr 234	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
37	36	adantrr 717	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
38		unieq 4878	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
39	38, 30	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = 𝑌)
40	39	eqeq2d 2740	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = 𝑌))
41		rabeq 3417	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
42	41	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ({𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
43	42	anbi2d 630	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
44	43	rexbidv 3157	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
45	44	ralbidv 3156	. . . . . . 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 632	. . . . . 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 3640	. . . . 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 801	. . . 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 821	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))))
51		3anass 1094	. 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 1086   = wceq 1540   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3402  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  ∪ cuni 4867  ‘cfv 6499  Fincfn 8895  Topctop 22756  LocFinclocfin 23367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-top 22757  df-locfin 23370

This theorem is referenced by:  finlocfin  23383  locfintop  23384  locfinbas  23385  locfinnei  23386  lfinun  23388  dissnlocfin  23392  locfindis  23393  locfincf  23394  locfinreflem  33803  locfinref  33804