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Theorem locfinnei 20475
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 2420 . . . 4  |-  U. A  =  U. A
31, 2islocfin 20469 . . 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 1022 . 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 2492 . . . . 5  |-  ( x  =  P  ->  (
x  e.  n  <->  P  e.  n ) )
65anbi1d 709 . . . 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 2937 . . 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 3178 . 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 473 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   E.wrex 2774   {crab 2777    i^i cin 3432   (/)c0 3758   U.cuni 4213   ` cfv 5592   Fincfn 7568   Topctop 19854   LocFinclocfin 20456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fv 5600  df-top 19858  df-locfin 20459
This theorem is referenced by:  lfinpfin  20476  lfinun  20477  locfincmp  20478  locfincf  20483
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