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Theorem elopabi 6873
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 4965 . . . 4  |-  Rel  { <. x ,  y >.  |  ph }
2 1st2nd 6858 . . . 4  |-  ( ( Rel  { <. x ,  y >.  |  ph }  /\  A  e.  { <. x ,  y >.  |  ph } )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
31, 2mpan 684 . . 3  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
4 id 22 . . 3  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  e.  { <. x ,  y >.  |  ph } )
53, 4eqeltrrd 2550 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph } )
6 fvex 5889 . . 3  |-  ( 1st `  A )  e.  _V
7 fvex 5889 . . 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 4723 . 2  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
115, 10sylib 201 1  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   <.cop 3965   {copab 4453   Rel wrel 4844   ` cfv 5589   1stc1st 6810   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-1st 6812  df-2nd 6813
This theorem is referenced by:  uhgrac  25111  wlkelwrd  25337  drngoi  26216  vci  26248  dicelval1sta  34826
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