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Theorem elopabi 6351
Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
Hypotheses
Ref Expression
elopabi.1  |-  ( x  =  ( 1st `  A
)  ->  ( ph  <->  ps ) )
elopabi.2  |-  ( y  =  ( 2nd `  A
)  ->  ( ps  <->  ch ) )
Assertion
Ref Expression
elopabi  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  ch )
Distinct variable groups:    x, y, A    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem elopabi
StepHypRef Expression
1 relopab 4941 . . . 4  |-  Rel  { <. x ,  y >.  |  ph }
2 1st2nd 6332 . . . 4  |-  ( ( Rel  { <. x ,  y >.  |  ph }  /\  A  e.  { <. x ,  y >.  |  ph } )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
31, 2mpan 652 . . 3  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
4 id 20 . . 3  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  e.  { <. x ,  y >.  |  ph } )
53, 4eqeltrrd 2462 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph } )
6 fvex 5682 . . 3  |-  ( 1st `  A )  e.  _V
7 fvex 5682 . . 3  |-  ( 2nd `  A )  e.  _V
8 elopabi.1 . . 3  |-  ( x  =  ( 1st `  A
)  ->  ( ph  <->  ps ) )
9 elopabi.2 . . 3  |-  ( y  =  ( 2nd `  A
)  ->  ( ps  <->  ch ) )
106, 7, 8, 9opelopab 4417 . 2  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
115, 10sylib 189 1  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   <.cop 3760   {copab 4206   Rel wrel 4823   ` cfv 5394   1stc1st 6286   2ndc2nd 6287
This theorem is referenced by:  drngoi  21843  vci  21875  dicelval1sta  31302
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fv 5402  df-1st 6288  df-2nd 6289
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