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Mirrors > Home > MPE Home > Th. List > locfinnei | Structured version Visualization version GIF version |
Description: A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Ref | Expression |
---|---|
locfinnei.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
locfinnei | β’ ((π΄ β (LocFinβπ½) β§ π β π) β βπ β π½ (π β π β§ {π β π΄ β£ (π β© π) β β } β Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | locfinnei.1 | . . . 4 β’ π = βͺ π½ | |
2 | eqid 2732 | . . . 4 β’ βͺ π΄ = βͺ π΄ | |
3 | 1, 2 | islocfin 23020 | . . 3 β’ (π΄ β (LocFinβπ½) β (π½ β Top β§ π = βͺ π΄ β§ βπ₯ β π βπ β π½ (π₯ β π β§ {π β π΄ β£ (π β© π) β β } β Fin))) |
4 | 3 | simp3bi 1147 | . 2 β’ (π΄ β (LocFinβπ½) β βπ₯ β π βπ β π½ (π₯ β π β§ {π β π΄ β£ (π β© π) β β } β Fin)) |
5 | eleq1 2821 | . . . . 5 β’ (π₯ = π β (π₯ β π β π β π)) | |
6 | 5 | anbi1d 630 | . . . 4 β’ (π₯ = π β ((π₯ β π β§ {π β π΄ β£ (π β© π) β β } β Fin) β (π β π β§ {π β π΄ β£ (π β© π) β β } β Fin))) |
7 | 6 | rexbidv 3178 | . . 3 β’ (π₯ = π β (βπ β π½ (π₯ β π β§ {π β π΄ β£ (π β© π) β β } β Fin) β βπ β π½ (π β π β§ {π β π΄ β£ (π β© π) β β } β Fin))) |
8 | 7 | rspccva 3611 | . 2 β’ ((βπ₯ β π βπ β π½ (π₯ β π β§ {π β π΄ β£ (π β© π) β β } β Fin) β§ π β π) β βπ β π½ (π β π β§ {π β π΄ β£ (π β© π) β β } β Fin)) |
9 | 4, 8 | sylan 580 | 1 β’ ((π΄ β (LocFinβπ½) β§ π β π) β βπ β π½ (π β π β§ {π β π΄ β£ (π β© π) β β } β Fin)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 βwrex 3070 {crab 3432 β© cin 3947 β c0 4322 βͺ cuni 4908 βcfv 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: lfinpfin 23027 lfinun 23028 locfincmp 23029 locfincf 23034 |
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