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Theorem isptfin 29754
Description: The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypothesis
Ref Expression
isptfin.1  |-  X  = 
U. A
Assertion
Ref Expression
isptfin  |-  ( A  e.  B  ->  ( A  e.  PtFin  <->  A. x  e.  X  { y  e.  A  |  x  e.  y }  e.  Fin ) )
Distinct variable groups:    x, y, A    x, X
Allowed substitution hints:    B( x, y)    X( y)

Proof of Theorem isptfin
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 unieq 4246 . . . 4  |-  ( a  =  A  ->  U. a  =  U. A )
2 isptfin.1 . . . 4  |-  X  = 
U. A
31, 2syl6eqr 2519 . . 3  |-  ( a  =  A  ->  U. a  =  X )
4 rabeq 3100 . . . 4  |-  ( a  =  A  ->  { y  e.  a  |  x  e.  y }  =  { y  e.  A  |  x  e.  y } )
54eleq1d 2529 . . 3  |-  ( a  =  A  ->  ( { y  e.  a  |  x  e.  y }  e.  Fin  <->  { y  e.  A  |  x  e.  y }  e.  Fin ) )
63, 5raleqbidv 3065 . 2  |-  ( a  =  A  ->  ( A. x  e.  U. a { y  e.  a  |  x  e.  y }  e.  Fin  <->  A. x  e.  X  { y  e.  A  |  x  e.  y }  e.  Fin ) )
7 df-ptfin 29724 . 2  |-  PtFin  =  {
a  |  A. x  e.  U. a { y  e.  a  |  x  e.  y }  e.  Fin }
86, 7elab2g 3245 1  |-  ( A  e.  B  ->  ( A  e.  PtFin  <->  A. x  e.  X  { y  e.  A  |  x  e.  y }  e.  Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   U.cuni 4238   Fincfn 7506   PtFincptfin 29720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-uni 4239  df-ptfin 29724
This theorem is referenced by:  finptfin  29756  ptfinfin  29757  lfinpfin  29762
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