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Theorem locfinnei 19852
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 2467 . . . 4  |-  U. A  =  U. A
31, 2islocfin 19846 . . 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 1013 . 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 2539 . . . . 5  |-  ( x  =  P  ->  (
x  e.  n  <->  P  e.  n ) )
65anbi1d 704 . . . 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 2973 . . 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 3213 . 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 471 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 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    i^i cin 3475   (/)c0 3785   U.cuni 4245   ` cfv 5588   Fincfn 7517   Topctop 19201   LocFinclocfin 19812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-top 19206  df-locfin 19816
This theorem is referenced by:  lfinpfin  19853  locfincmp  19854  locfincf  19859
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