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Theorem ptfinfin 20186
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 20183 . . . 4  |-  ( A  e.  PtFin  ->  ( A  e.  PtFin 
<-> 
A. p  e.  X  { x  e.  A  |  p  e.  x }  e.  Fin )
)
32ibi 241 . . 3  |-  ( A  e.  PtFin  ->  A. p  e.  X  { x  e.  A  |  p  e.  x }  e.  Fin )
4 eleq1 2526 . . . . . 6  |-  ( p  =  P  ->  (
p  e.  x  <->  P  e.  x ) )
54rabbidv 3098 . . . . 5  |-  ( p  =  P  ->  { x  e.  A  |  p  e.  x }  =  {
x  e.  A  |  P  e.  x }
)
65eleq1d 2523 . . . 4  |-  ( p  =  P  ->  ( { x  e.  A  |  p  e.  x }  e.  Fin  <->  { x  e.  A  |  P  e.  x }  e.  Fin ) )
76rspccv 3204 . . 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 427 1  |-  ( ( A  e.  PtFin  /\  P  e.  X )  ->  { x  e.  A  |  P  e.  x }  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   U.cuni 4235   Fincfn 7509   PtFincptfin 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-uni 4236  df-ptfin 20173
This theorem is referenced by:  locfindis  20197  comppfsc  20199
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