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Theorem isptfin 26059
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 3959 . . . 4  |-  ( a  =  A  ->  U. a  =  U. A )
2 isptfin.1 . . . 4  |-  X  = 
U. A
31, 2syl6eqr 2430 . . 3  |-  ( a  =  A  ->  U. a  =  X )
4 rabeq 2886 . . . 4  |-  ( a  =  A  ->  { y  e.  a  |  x  e.  y }  =  { y  e.  A  |  x  e.  y } )
54eleq1d 2446 . . 3  |-  ( a  =  A  ->  ( { y  e.  a  |  x  e.  y }  e.  Fin  <->  { y  e.  A  |  x  e.  y }  e.  Fin ) )
63, 5raleqbidv 2852 . 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 26029 . 2  |-  PtFin  =  {
a  |  A. x  e.  U. a { y  e.  a  |  x  e.  y }  e.  Fin }
86, 7elab2g 3020 1  |-  ( A  e.  B  ->  ( A  e.  PtFin  <->  A. x  e.  X  { y  e.  A  |  x  e.  y }  e.  Fin ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646   U.cuni 3950   Fincfn 7038   PtFincptfin 26025
This theorem is referenced by:  finptfin  26061  ptfinfin  26062  lfinpfin  26067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-uni 3951  df-ptfin 26029
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