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

Theorem locfinnei 26302
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 2283 . . . 4  |-  U. A  =  U. A
31, 2islocfin 26296 . . 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 972 . 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 2343 . . . . 5  |-  ( x  =  P  ->  (
x  e.  n  <->  P  e.  n ) )
65anbi1d 685 . . . 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 2564 . . 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 2883 . 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 457 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151   (/)c0 3455   U.cuni 3827   ` cfv 5255   Fincfn 6863   Topctop 16631   LocFinclocfin 26262
This theorem is referenced by:  lfinpfin  26303  locfincmp  26304  locfincf  26306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-locfin 26266
  Copyright terms: Public domain W3C validator