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Theorem locfinnei 26074
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 2388 . . . 4  |-  U. A  =  U. A
31, 2islocfin 26068 . . 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 2448 . . . . 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 2671 . . 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 2995 . 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 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   {crab 2654    i^i cin 3263   (/)c0 3572   U.cuni 3958   ` cfv 5395   Fincfn 7046   Topctop 16882   LocFinclocfin 26034
This theorem is referenced by:  lfinpfin  26075  locfincmp  26076  locfincf  26078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fv 5403  df-top 16887  df-locfin 26038
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