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Theorem isptfin 26398
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 3852 . . . 4  |-  ( a  =  A  ->  U. a  =  U. A )
2 isptfin.1 . . . 4  |-  X  = 
U. A
31, 2syl6eqr 2346 . . 3  |-  ( a  =  A  ->  U. a  =  X )
4 rabeq 2795 . . . 4  |-  ( a  =  A  ->  { y  e.  a  |  x  e.  y }  =  { y  e.  A  |  x  e.  y } )
54eleq1d 2362 . . 3  |-  ( a  =  A  ->  ( { y  e.  a  |  x  e.  y }  e.  Fin  <->  { y  e.  A  |  x  e.  y }  e.  Fin ) )
63, 5raleqbidv 2761 . 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 26368 . 2  |-  PtFin  =  {
a  |  A. x  e.  U. a { y  e.  a  |  x  e.  y }  e.  Fin }
86, 7elab2g 2929 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 176    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   U.cuni 3843   Fincfn 6879   PtFincptfin 26364
This theorem is referenced by:  finptfin  26400  ptfinfin  26401  lfinpfin  26406
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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