Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  locfinnei Structured version   Unicode version

Theorem locfinnei 26373
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.)
Hypothesis
Ref Expression
locfinnei.1  |-  X  = 
U. J
Assertion
Ref Expression
locfinnei  |-  ( ( A  e.  ( LocFin `  J )  /\  P  e.  X )  ->  E. n  e.  J  ( P  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
Distinct variable groups:    n, s, A    n, J    P, n
Allowed substitution hints:    P( s)    J( s)    X( n, s)

Proof of Theorem locfinnei
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 locfinnei.1 . . . 4  |-  X  = 
U. J
2 eqid 2435 . . . 4  |-  U. A  =  U. A
31, 2islocfin 26367 . . 3  |-  ( A  e.  ( LocFin `  J
)  <->  ( J  e. 
Top  /\  X  =  U. A  /\  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
43simp3bi 974 . 2  |-  ( A  e.  ( LocFin `  J
)  ->  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
5 eleq1 2495 . . . . 5  |-  ( x  =  P  ->  (
x  e.  n  <->  P  e.  n ) )
65anbi1d 686 . . . 4  |-  ( x  =  P  ->  (
( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  <->  ( P  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
76rexbidv 2718 . . 3  |-  ( x  =  P  ->  ( E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  <->  E. n  e.  J  ( P  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
87rspccva 3043 . 2  |-  ( ( A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  /\  P  e.  X )  ->  E. n  e.  J  ( P  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
94, 8sylan 458 1  |-  ( ( A  e.  ( LocFin `  J )  /\  P  e.  X )  ->  E. n  e.  J  ( P  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    i^i cin 3311   (/)c0 3620   U.cuni 4007   ` cfv 5446   Fincfn 7101   Topctop 16950   LocFinclocfin 26333
This theorem is referenced by:  lfinpfin  26374  locfincmp  26375  locfincf  26377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-top 16955  df-locfin 26337
  Copyright terms: Public domain W3C validator