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Theorem ptfinfin 26298
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 26295 . . . 4  |-  ( A  e.  PtFin  ->  ( A  e.  PtFin 
<-> 
A. p  e.  X  { x  e.  A  |  p  e.  x }  e.  Fin )
)
32ibi 232 . . 3  |-  ( A  e.  PtFin  ->  A. p  e.  X  { x  e.  A  |  p  e.  x }  e.  Fin )
4 eleq1 2343 . . . . . 6  |-  ( p  =  P  ->  (
p  e.  x  <->  P  e.  x ) )
54rabbidv 2780 . . . . 5  |-  ( p  =  P  ->  { x  e.  A  |  p  e.  x }  =  {
x  e.  A  |  P  e.  x }
)
65eleq1d 2349 . . . 4  |-  ( p  =  P  ->  ( { x  e.  A  |  p  e.  x }  e.  Fin  <->  { x  e.  A  |  P  e.  x }  e.  Fin ) )
76rspccv 2881 . . 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 15 . 2  |-  ( A  e.  PtFin  ->  ( P  e.  X  ->  { x  e.  A  |  P  e.  x }  e.  Fin ) )
98imp 418 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   U.cuni 3827   Fincfn 6863   PtFincptfin 26261
This theorem is referenced by:  locfindis  26305  comppfsc  26307
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-uni 3828  df-ptfin 26265
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