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Theorem locfincmp 23250
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 23247 . . . . . . . . 9 ((𝐢 ∈ (LocFinβ€˜π½) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin))
32ralrimiva 3144 . . . . . . . 8 (𝐢 ∈ (LocFinβ€˜π½) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ π‘œ) β‰  βˆ…} ∈ Fin))
41cmpcov2 23114 . . . . . . . 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 591 . . . . . . 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 9356 . . . . . . . . 9 (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin))
7 simplrr 774 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) β†’ 𝑐 ∈ Fin)
8 eldifsn 4789 . . . . . . . . . . . . 13 (π‘₯ ∈ (𝐢 βˆ– {βˆ…}) ↔ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  βˆ…))
9 ineq1 4204 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = π‘₯ β†’ (𝑠 ∩ π‘œ) = (π‘₯ ∩ π‘œ))
109neeq1d 2998 . . . . . . . . . . . . . . . . . . 19 (𝑠 = π‘₯ β†’ ((𝑠 ∩ π‘œ) β‰  βˆ… ↔ (π‘₯ ∩ π‘œ) β‰  βˆ…))
11 simplrl 773 . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . 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 4463 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ π‘₯ ∧ 𝑦 ∈ π‘œ) β†’ (π‘₯ ∩ π‘œ) β‰  βˆ…)
1512, 13, 14syl2anc 582 . . . . . . . . . . . . . . . . . . 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 3684 . . . . . . . . . . . . . . . . . 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 4912 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ π‘₯ ∧ π‘₯ ∈ 𝐢) β†’ 𝑦 ∈ βˆͺ 𝐢)
18 locfincmp.2 . . . . . . . . . . . . . . . . . . . . . . . 24 π‘Œ = βˆͺ 𝐢
1917, 18eleqtrrdi 2842 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ π‘₯ ∧ π‘₯ ∈ 𝐢) β†’ 𝑦 ∈ π‘Œ)
2019ancoms 457 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ π‘Œ)
2120adantl 480 . . . . . . . . . . . . . . . . . . . . 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 23246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐢 ∈ (LocFinβ€˜π½) β†’ 𝑋 = π‘Œ)
2322adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ 𝑋 = π‘Œ)
2423ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) ∧ (𝑐 βŠ† 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = βˆͺ 𝑐) ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ π‘₯)) β†’ 𝑦 ∈ 𝑋)
26 simplr 765 . . . . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . . . 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 4911 . . . . . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . . . 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 1934 . . . . . . . . . . . . . . 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 4345 . . . . . . . . . . . . . . 15 (π‘₯ β‰  βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ π‘₯)
34 eliun 5000 . . . . . . . . . . . . . . 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 295 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 3987 . . . . . . . . . . 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 9342 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 9175 . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 506 . . . . . . . . . . 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 582 . . . . . . . . . 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 452 . . . . . . . . 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 592 . . . . . . . 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 3153 . . . . . . 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 9046 . . . . . 6 {βˆ…} ∈ Fin
50 unfi 9174 . . . . . 6 (((𝐢 βˆ– {βˆ…}) ∈ Fin ∧ {βˆ…} ∈ Fin) β†’ ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…}) ∈ Fin)
5148, 49, 50sylancl 584 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…}) ∈ Fin)
52 ssun1 4171 . . . . . 6 𝐢 βŠ† (𝐢 βˆͺ {βˆ…})
53 undif1 4474 . . . . . 6 ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…}) = (𝐢 βˆͺ {βˆ…})
5452, 53sseqtrri 4018 . . . . 5 𝐢 βŠ† ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…})
55 ssfi 9175 . . . . 5 ((((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…}) ∈ Fin ∧ 𝐢 βŠ† ((𝐢 βˆ– {βˆ…}) βˆͺ {βˆ…})) β†’ 𝐢 ∈ Fin)
5651, 54, 55sylancl 584 . . . 4 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ 𝐢 ∈ Fin)
5756, 23jca 510 . . 3 ((𝐽 ∈ Comp ∧ 𝐢 ∈ (LocFinβ€˜π½)) β†’ (𝐢 ∈ Fin ∧ 𝑋 = π‘Œ))
5857ex 411 . 2 (𝐽 ∈ Comp β†’ (𝐢 ∈ (LocFinβ€˜π½) β†’ (𝐢 ∈ Fin ∧ 𝑋 = π‘Œ)))
59 cmptop 23119 . . 3 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
601, 18finlocfin 23244 . . . 4 ((𝐽 ∈ Top ∧ 𝐢 ∈ Fin ∧ 𝑋 = π‘Œ) β†’ 𝐢 ∈ (LocFinβ€˜π½))
61603expib 1120 . . 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 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430   βˆ– cdif 3944   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907  βˆͺ ciun 4996  β€˜cfv 6542  Fincfn 8941  Topctop 22615  Compccmp 23110  LocFinclocfin 23228
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-en 8942  df-fin 8945  df-top 22616  df-cmp 23111  df-locfin 23231
This theorem is referenced by: (None)
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