Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isptfin Unicode version

Theorem isptfin 26295
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 3836 . . . 4  |-  ( a  =  A  ->  U. a  =  U. A )
2 isptfin.1 . . . 4  |-  X  = 
U. A
31, 2syl6eqr 2333 . . 3  |-  ( a  =  A  ->  U. a  =  X )
4 rabeq 2782 . . . 4  |-  ( a  =  A  ->  { y  e.  a  |  x  e.  y }  =  { y  e.  A  |  x  e.  y } )
54eleq1d 2349 . . 3  |-  ( a  =  A  ->  ( { y  e.  a  |  x  e.  y }  e.  Fin  <->  { y  e.  A  |  x  e.  y }  e.  Fin ) )
63, 5raleqbidv 2748 . 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 26265 . 2  |-  PtFin  =  {
a  |  A. x  e.  U. a { y  e.  a  |  x  e.  y }  e.  Fin }
86, 7elab2g 2916 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   U.cuni 3827   Fincfn 6863   PtFincptfin 26261
This theorem is referenced by:  finptfin  26297  ptfinfin  26298  lfinpfin  26303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-uni 3828  df-ptfin 26265
  Copyright terms: Public domain W3C validator