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Theorem ptfinfin 26380
Description: A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypothesis
Ref Expression
ptfinfin.1  |-  X  = 
U. A
Assertion
Ref Expression
ptfinfin  |-  ( ( A  e.  PtFin  /\  P  e.  X )  ->  { x  e.  A  |  P  e.  x }  e.  Fin )
Distinct variable groups:    x, A    x, P    x, X

Proof of Theorem ptfinfin
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 ptfinfin.1 . . . . 5  |-  X  = 
U. A
21isptfin 26377 . . . 4  |-  ( A  e.  PtFin  ->  ( A  e.  PtFin 
<-> 
A. p  e.  X  { x  e.  A  |  p  e.  x }  e.  Fin )
)
32ibi 234 . . 3  |-  ( A  e.  PtFin  ->  A. p  e.  X  { x  e.  A  |  p  e.  x }  e.  Fin )
4 eleq1 2498 . . . . . 6  |-  ( p  =  P  ->  (
p  e.  x  <->  P  e.  x ) )
54rabbidv 2950 . . . . 5  |-  ( p  =  P  ->  { x  e.  A  |  p  e.  x }  =  {
x  e.  A  |  P  e.  x }
)
65eleq1d 2504 . . . 4  |-  ( p  =  P  ->  ( { x  e.  A  |  p  e.  x }  e.  Fin  <->  { x  e.  A  |  P  e.  x }  e.  Fin ) )
76rspccv 3051 . . 3  |-  ( A. p  e.  X  {
x  e.  A  |  p  e.  x }  e.  Fin  ->  ( P  e.  X  ->  { x  e.  A  |  P  e.  x }  e.  Fin ) )
83, 7syl 16 . 2  |-  ( A  e.  PtFin  ->  ( P  e.  X  ->  { x  e.  A  |  P  e.  x }  e.  Fin ) )
98imp 420 1  |-  ( ( A  e.  PtFin  /\  P  e.  X )  ->  { x  e.  A  |  P  e.  x }  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   U.cuni 4017   Fincfn 7111   PtFincptfin 26343
This theorem is referenced by:  locfindis  26387  comppfsc  26389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-uni 4018  df-ptfin 26347
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