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Theorem islocfin 23020
Description: The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypotheses
Ref Expression
islocfin.1 𝑋 = βˆͺ 𝐽
islocfin.2 π‘Œ = βˆͺ 𝐴
Assertion
Ref Expression
islocfin (𝐴 ∈ (LocFinβ€˜π½) ↔ (𝐽 ∈ Top ∧ 𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
Distinct variable groups:   𝑛,𝑠,π‘₯,𝐴   𝑛,𝐽,π‘₯   π‘₯,𝑋
Allowed substitution hints:   𝐽(𝑠)   𝑋(𝑛,𝑠)   π‘Œ(π‘₯,𝑛,𝑠)

Proof of Theorem islocfin
Dummy variables 𝑗 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-locfin 23010 . . . 4 LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ (βˆͺ 𝑗 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ βˆͺ π‘—βˆƒπ‘› ∈ 𝑗 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))})
21mptrcl 7007 . . 3 (𝐴 ∈ (LocFinβ€˜π½) β†’ 𝐽 ∈ Top)
3 eqimss2 4041 . . . . . . . . . . 11 (𝑋 = βˆͺ 𝑦 β†’ βˆͺ 𝑦 βŠ† 𝑋)
4 sspwuni 5103 . . . . . . . . . . 11 (𝑦 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝑦 βŠ† 𝑋)
53, 4sylibr 233 . . . . . . . . . 10 (𝑋 = βˆͺ 𝑦 β†’ 𝑦 βŠ† 𝒫 𝑋)
6 velpw 4607 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝒫 𝑋 ↔ 𝑦 βŠ† 𝒫 𝑋)
75, 6sylibr 233 . . . . . . . . 9 (𝑋 = βˆͺ 𝑦 β†’ 𝑦 ∈ 𝒫 𝒫 𝑋)
87adantr 481 . . . . . . . 8 ((𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)) β†’ 𝑦 ∈ 𝒫 𝒫 𝑋)
98abssi 4067 . . . . . . 7 {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} βŠ† 𝒫 𝒫 𝑋
10 islocfin.1 . . . . . . . . 9 𝑋 = βˆͺ 𝐽
1110topopn 22407 . . . . . . . 8 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
12 pwexg 5376 . . . . . . . 8 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
13 pwexg 5376 . . . . . . . 8 (𝒫 𝑋 ∈ V β†’ 𝒫 𝒫 𝑋 ∈ V)
1411, 12, 133syl 18 . . . . . . 7 (𝐽 ∈ Top β†’ 𝒫 𝒫 𝑋 ∈ V)
15 ssexg 5323 . . . . . . 7 (({𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} βŠ† 𝒫 𝒫 𝑋 ∧ 𝒫 𝒫 𝑋 ∈ V) β†’ {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} ∈ V)
169, 14, 15sylancr 587 . . . . . 6 (𝐽 ∈ Top β†’ {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} ∈ V)
17 unieq 4919 . . . . . . . . . . 11 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
1817, 10eqtr4di 2790 . . . . . . . . . 10 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
1918eqeq1d 2734 . . . . . . . . 9 (𝑗 = 𝐽 β†’ (βˆͺ 𝑗 = βˆͺ 𝑦 ↔ 𝑋 = βˆͺ 𝑦))
20 rexeq 3321 . . . . . . . . . 10 (𝑗 = 𝐽 β†’ (βˆƒπ‘› ∈ 𝑗 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
2118, 20raleqbidv 3342 . . . . . . . . 9 (𝑗 = 𝐽 β†’ (βˆ€π‘₯ ∈ βˆͺ π‘—βˆƒπ‘› ∈ 𝑗 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
2219, 21anbi12d 631 . . . . . . . 8 (𝑗 = 𝐽 β†’ ((βˆͺ 𝑗 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ βˆͺ π‘—βˆƒπ‘› ∈ 𝑗 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)) ↔ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))))
2322abbidv 2801 . . . . . . 7 (𝑗 = 𝐽 β†’ {𝑦 ∣ (βˆͺ 𝑗 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ βˆͺ π‘—βˆƒπ‘› ∈ 𝑗 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} = {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))})
2423, 1fvmptg 6996 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} ∈ V) β†’ (LocFinβ€˜π½) = {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))})
2516, 24mpdan 685 . . . . 5 (𝐽 ∈ Top β†’ (LocFinβ€˜π½) = {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))})
2625eleq2d 2819 . . . 4 (𝐽 ∈ Top β†’ (𝐴 ∈ (LocFinβ€˜π½) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))}))
27 elex 3492 . . . . . 6 (𝐴 ∈ {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} β†’ 𝐴 ∈ V)
2827adantl 482 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))}) β†’ 𝐴 ∈ V)
29 simpr 485 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 = π‘Œ) β†’ 𝑋 = π‘Œ)
30 islocfin.2 . . . . . . . . . 10 π‘Œ = βˆͺ 𝐴
3129, 30eqtrdi 2788 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = π‘Œ) β†’ 𝑋 = βˆͺ 𝐴)
3211adantr 481 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = π‘Œ) β†’ 𝑋 ∈ 𝐽)
3331, 32eqeltrrd 2834 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = π‘Œ) β†’ βˆͺ 𝐴 ∈ 𝐽)
3433elexd 3494 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = π‘Œ) β†’ βˆͺ 𝐴 ∈ V)
35 uniexb 7750 . . . . . . 7 (𝐴 ∈ V ↔ βˆͺ 𝐴 ∈ V)
3634, 35sylibr 233 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = π‘Œ) β†’ 𝐴 ∈ V)
3736adantrr 715 . . . . 5 ((𝐽 ∈ Top ∧ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))) β†’ 𝐴 ∈ V)
38 unieq 4919 . . . . . . . . 9 (𝑦 = 𝐴 β†’ βˆͺ 𝑦 = βˆͺ 𝐴)
3938, 30eqtr4di 2790 . . . . . . . 8 (𝑦 = 𝐴 β†’ βˆͺ 𝑦 = π‘Œ)
4039eqeq2d 2743 . . . . . . 7 (𝑦 = 𝐴 β†’ (𝑋 = βˆͺ 𝑦 ↔ 𝑋 = π‘Œ))
41 rabeq 3446 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…})
4241eleq1d 2818 . . . . . . . . . 10 (𝑦 = 𝐴 β†’ ({𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
4342anbi2d 629 . . . . . . . . 9 (𝑦 = 𝐴 β†’ ((π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
4443rexbidv 3178 . . . . . . . 8 (𝑦 = 𝐴 β†’ (βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
4544ralbidv 3177 . . . . . . 7 (𝑦 = 𝐴 β†’ (βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
4640, 45anbi12d 631 . . . . . 6 (𝑦 = 𝐴 β†’ ((𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)) ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))))
4746elabg 3666 . . . . 5 (𝐴 ∈ V β†’ (𝐴 ∈ {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))))
4828, 37, 47pm5.21nd 800 . . . 4 (𝐽 ∈ Top β†’ (𝐴 ∈ {𝑦 ∣ (𝑋 = βˆͺ 𝑦 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))} ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))))
4926, 48bitrd 278 . . 3 (𝐽 ∈ Top β†’ (𝐴 ∈ (LocFinβ€˜π½) ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))))
502, 49biadanii 820 . 2 (𝐴 ∈ (LocFinβ€˜π½) ↔ (𝐽 ∈ Top ∧ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))))
51 3anass 1095 . 2 ((𝐽 ∈ Top ∧ 𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)) ↔ (𝐽 ∈ Top ∧ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))))
5250, 51bitr4i 277 1 (𝐴 ∈ (LocFinβ€˜π½) ↔ (𝐽 ∈ Top ∧ 𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908  β€˜cfv 6543  Fincfn 8938  Topctop 22394  LocFinclocfin 23007
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22395  df-locfin 23010
This theorem is referenced by:  finlocfin  23023  locfintop  23024  locfinbas  23025  locfinnei  23026  lfinun  23028  dissnlocfin  23032  locfindis  23033  locfincf  23034  locfinreflem  32815  locfinref  32816
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