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Theorem locfincmp 23030
Description: For a compact space, the locally finite covers are precisely the finite covers. Sadly, this property does not properly characterize all compact spaces. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
locfincmp.1 𝑋 = βˆͺ 𝐽
locfincmp.2 π‘Œ = βˆͺ 𝐢
Assertion
Ref Expression
locfincmp (𝐽 ∈ Comp β†’ (𝐢 ∈ (LocFinβ€˜π½) ↔ (𝐢 ∈ Fin ∧ 𝑋 = π‘Œ)))
Proof of Theorem locfincmp
Dummy variables π‘œ 𝑐 𝑠 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 locfincmp.1 . . . . . . . . . 10 𝑋 = βˆͺ 𝐽
21locfinnei 23027 . . . . . . . . 9 ((𝐢 ∈ (LocFinβ€˜π½) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin))
32ralrimiva 3147 . . . . . . . 8 (𝐢 ∈ (LocFinβ€˜π½) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin))
41cmpcov2 22894 . . . . . . . 8 ((𝐽 ∈ Comp ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin)) β†’ βˆƒπ‘ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑐 ∧ βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin))
53, 4sylan2 594 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ βˆƒπ‘ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑐 ∧ βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin))
6 elfpw 9354 . . . . . . . . 9 (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin))
7 simplrr 777 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) β†’ 𝑐 ∈ Fin)
8 eldifsn 4791 . . . . . . . . . . . . 13 (π‘₯ ∈ (𝐢 βˆ– {βˆ…}) ↔ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  βˆ…))
9 ineq1 4206 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = π‘₯ β†’ (𝑠 ∩ π‘œ) = (π‘₯ ∩ π‘œ))
109neeq1d 3001 . . . . . . . . . . . . . . . . . . 19 (𝑠 = π‘₯ β†’ ((𝑠 ∩ π‘œ) β‰  βˆ… ↔ (π‘₯ ∩ π‘œ) β‰  βˆ…))
11 simplrl 776 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) ∧ (π‘œ ∈ 𝑐 ∧ 𝑦 ∈ π‘œ)) β†’ π‘₯ ∈ 𝐢)
12 simplrr 777 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) ∧ (π‘œ ∈ 𝑐 ∧ 𝑦 ∈ π‘œ)) β†’ 𝑦 ∈ π‘₯)
13 simprr 772 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) ∧ (π‘œ ∈ 𝑐 ∧ 𝑦 ∈ π‘œ)) β†’ 𝑦 ∈ π‘œ)
14 inelcm 4465 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ π‘₯ ∧ 𝑦 ∈ π‘œ) β†’ (π‘₯ ∩ π‘œ) β‰  βˆ…)
1512, 13, 14syl2anc 585 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) ∧ (π‘œ ∈ 𝑐 ∧ 𝑦 ∈ π‘œ)) β†’ (π‘₯ ∩ π‘œ) β‰  βˆ…)
1610, 11, 15elrabd 3686 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) ∧ (π‘œ ∈ 𝑐 ∧ 𝑦 ∈ π‘œ)) β†’ π‘₯ ∈ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…})
17 elunii 4914 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ π‘₯ ∧ π‘₯ ∈ 𝐢) β†’ 𝑦 ∈ βˆͺ 𝐢)
18 locfincmp.2 . . . . . . . . . . . . . . . . . . . . . . . 24 π‘Œ = βˆͺ 𝐢
1917, 18eleqtrrdi 2845 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ π‘₯ ∧ π‘₯ ∈ 𝐢) β†’ 𝑦 ∈ π‘Œ)
2019ancoms 460 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ π‘Œ)
2120adantl 483 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) β†’ 𝑦 ∈ π‘Œ)
221, 18locfinbas 23026 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐢 ∈ (LocFinβ€˜π½) β†’ 𝑋 = π‘Œ)
2322adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ 𝑋 = π‘Œ)
2423ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) β†’ 𝑋 = π‘Œ)
2521, 24eleqtrrd 2837 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) β†’ 𝑦 ∈ 𝑋)
26 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) β†’ 𝑋 = βˆͺ 𝑐)
2725, 26eleqtrd 2836 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) β†’ 𝑦 ∈ βˆͺ 𝑐)
28 eluni2 4913 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ βˆͺ 𝑐 ↔ βˆƒπ‘œ ∈ 𝑐 𝑦 ∈ π‘œ)
2927, 28sylib 217 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) β†’ βˆƒπ‘œ ∈ 𝑐 𝑦 ∈ π‘œ)
3016, 29reximddv 3172 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) β†’ βˆƒπ‘œ ∈ 𝑐 π‘₯ ∈ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…})
3130expr 458 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ π‘₯ ∈ 𝐢) β†’ (𝑦 ∈ π‘₯ β†’ βˆƒπ‘œ ∈ 𝑐 π‘₯ ∈ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…}))
3231exlimdv 1937 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ π‘₯ ∈ 𝐢) β†’ (βˆƒπ‘¦ 𝑦 ∈ π‘₯ β†’ βˆƒπ‘œ ∈ 𝑐 π‘₯ ∈ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…}))
33 n0 4347 . . . . . . . . . . . . . . 15 (π‘₯ β‰  βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ π‘₯)
34 eliun 5002 . . . . . . . . . . . . . . 15 (π‘₯ ∈ βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ↔ βˆƒπ‘œ ∈ 𝑐 π‘₯ ∈ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…})
3532, 33, 343imtr4g 296 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ π‘₯ ∈ 𝐢) β†’ (π‘₯ β‰  βˆ… β†’ π‘₯ ∈ βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…}))
3635expimpd 455 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) β†’ ((π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ ∈ βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…}))
378, 36biimtrid 241 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) β†’ (π‘₯ ∈ (𝐢 βˆ– {βˆ…}) β†’ π‘₯ ∈ βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…}))
3837ssrdv 3989 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) β†’ (𝐢 βˆ– {βˆ…}) βŠ† βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…})
39 iunfi 9340 . . . . . . . . . . . . 13 ((𝑐 ∈ Fin ∧ βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin) β†’ βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin)
4039ex 414 . . . . . . . . . . . 12 (𝑐 ∈ Fin β†’ (βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin β†’ βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin))
41 ssfi 9173 . . . . . . . . . . . . 13 ((βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin ∧ (𝐢 βˆ– {βˆ…}) βŠ† βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…}) β†’ (𝐢 βˆ– {βˆ…}) ∈ Fin)
4241expcom 415 . . . . . . . . . . . 12 ((𝐢 βˆ– {βˆ…}) βŠ† βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} β†’ (βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin β†’ (𝐢 βˆ– {βˆ…}) ∈ Fin))
4340, 42sylan9 509 . . . . . . . . . . 11 ((𝑐 ∈ Fin ∧ (𝐢 βˆ– {βˆ…}) βŠ† βˆͺ π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…}) β†’ (βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin β†’ (𝐢 βˆ– {βˆ…}) ∈ Fin))
447, 38, 43syl2anc 585 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) β†’ (βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin β†’ (𝐢 βˆ– {βˆ…}) ∈ Fin))
4544expimpd 455 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) β†’ ((𝑋 = βˆͺ 𝑐 ∧ βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin) β†’ (𝐢 βˆ– {βˆ…}) ∈ Fin))
466, 45sylan2b 595 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) β†’ ((𝑋 = βˆͺ 𝑐 ∧ βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin) β†’ (𝐢 βˆ– {βˆ…}) ∈ Fin))
4746rexlimdva 3156 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ (βˆƒπ‘ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑐 ∧ βˆ€π‘œ ∈ 𝑐 {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin) β†’ (𝐢 βˆ– {βˆ…}) ∈ Fin))
485, 47mpd 15 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ (𝐢 βˆ– {βˆ…}) ∈ Fin)
49 snfi 9044 . . . . . 6 {βˆ…} ∈ Fin
50 unfi 9172 . . . . . 6 (((𝐢 βˆ– {βˆ…}) ∈ Fin ∧ {βˆ…} ∈ Fin) β†’ ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…}) ∈ Fin)
5148, 49, 50sylancl 587 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…}) ∈ Fin)
52 ssun1 4173 . . . . . 6 𝐢 βŠ† (𝐢 βˆͺ {βˆ…})
53 undif1 4476 . . . . . 6 ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…}) = (𝐢 βˆͺ {βˆ…})
5452, 53sseqtrri 4020 . . . . 5 𝐢 βŠ† ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…})
55 ssfi 9173 . . . . 5 ((((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…}) ∈ Fin ∧ 𝐢 βŠ† ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…})) β†’ 𝐢 ∈ Fin)
5651, 54, 55sylancl 587 . . . 4 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ 𝐢 ∈ Fin)
5756, 23jca 513 . . 3 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ (𝐢 ∈ Fin ∧ 𝑋 = π‘Œ))
5857ex 414 . 2 (𝐽 ∈ Comp β†’ (𝐢 ∈ (LocFinβ€˜π½) β†’ (𝐢 ∈ Fin ∧ 𝑋 = π‘Œ)))
59 cmptop 22899 . . 3 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
601, 18finlocfin 23024 . . . 4 ((𝐽 ∈ Top ∧ 𝐢 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ 𝐢 ∈ (LocFinβ€˜π½))
61603expib 1123 . . 3 (𝐽 ∈ Top β†’ ((𝐢 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ 𝐢 ∈ (LocFinβ€˜π½)))
6259, 61syl 17 . 2 (𝐽 ∈ Comp β†’ ((𝐢 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ 𝐢 ∈ (LocFinβ€˜π½)))
6358, 62impbid 211 1 (𝐽 ∈ Comp β†’ (𝐢 ∈ (LocFinβ€˜π½) ↔ (𝐢 ∈ Fin ∧ 𝑋 = π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433   βˆ– cdif 3946   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909  βˆͺ ciun 4998  β€˜cfv 6544  Fincfn 8939  Topctop 22395  Compccmp 22890  LocFinclocfin 23008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-en 8940  df-fin 8943  df-top 22396  df-cmp 22891  df-locfin 23011
This theorem is referenced by: (None)
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