![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2726 | . . . 4 β’ βͺ π΄ = βͺ π΄ | |
3 | 1, 2 | islocfin 23376 | . . 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 1144 | . 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 2815 | . . . . 5 β’ (π₯ = π β (π₯ β π β π β π)) | |
6 | 5 | anbi1d 629 | . . . 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 3172 | . . 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 3605 | . 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 579 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 βwrex 3064 {crab 3426 β© cin 3942 β c0 4317 βͺ cuni 4902 βcfv 6537 Fincfn 8941 Topctop 22750 LocFinclocfin 23363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fv 6545 df-top 22751 df-locfin 23366 |
This theorem is referenced by: lfinpfin 23383 lfinun 23384 locfincmp 23385 locfincf 23390 |
Copyright terms: Public domain | W3C validator |