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Theorem isptfin 20143
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 4259 . . . 4  |-  ( a  =  A  ->  U. a  =  U. A )
2 isptfin.1 . . . 4  |-  X  = 
U. A
31, 2syl6eqr 2516 . . 3  |-  ( a  =  A  ->  U. a  =  X )
4 rabeq 3103 . . . 4  |-  ( a  =  A  ->  { y  e.  a  |  x  e.  y }  =  { y  e.  A  |  x  e.  y } )
54eleq1d 2526 . . 3  |-  ( a  =  A  ->  ( { y  e.  a  |  x  e.  y }  e.  Fin  <->  { y  e.  A  |  x  e.  y }  e.  Fin ) )
63, 5raleqbidv 3068 . 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 20133 . 2  |-  PtFin  =  {
a  |  A. x  e.  U. a { y  e.  a  |  x  e.  y }  e.  Fin }
86, 7elab2g 3248 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 1395    e. wcel 1819   A.wral 2807   {crab 2811   U.cuni 4251   Fincfn 7535   PtFincptfin 20130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-uni 4252  df-ptfin 20133
This theorem is referenced by:  finptfin  20145  ptfinfin  20146  lfinpfin  20151
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