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Theorem ptfinfin 26401
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 26398 . . . 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 2356 . . . . . 6  |-  ( p  =  P  ->  (
p  e.  x  <->  P  e.  x ) )
54rabbidv 2793 . . . . 5  |-  ( p  =  P  ->  { x  e.  A  |  p  e.  x }  =  {
x  e.  A  |  P  e.  x }
)
65eleq1d 2362 . . . 4  |-  ( p  =  P  ->  ( { x  e.  A  |  p  e.  x }  e.  Fin  <->  { x  e.  A  |  P  e.  x }  e.  Fin ) )
76rspccv 2894 . . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   U.cuni 3843   Fincfn 6879   PtFincptfin 26364
This theorem is referenced by:  locfindis  26408  comppfsc  26410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-uni 3844  df-ptfin 26368
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