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Theorem elopabi 5849
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 4492 . . . 4  |-  Rel  { <. x ,  y >.  |  ph }
2 1st2nd 5835 . . . 4  |-  ( ( Rel  { <. x ,  y >.  |  ph }  /\  A  e.  { <. x ,  y >.  |  ph } )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
31, 2mpan 408 . . 3  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
4 id 19 . . 3  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  e.  { <. x ,  y >.  |  ph } )
53, 4eqeltrrd 2131 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph } )
6 1stexg 5822 . . 3  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  ( 1st `  A
)  e.  _V )
7 2ndexg 5823 . . 3  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  ( 2nd `  A
)  e.  _V )
8 elopabi.1 . . . 4  |-  ( x  =  ( 1st `  A
)  ->  ( ph  <->  ps ) )
9 elopabi.2 . . . 4  |-  ( y  =  ( 2nd `  A
)  ->  ( ps  <->  ch ) )
108, 9opelopabg 4033 . . 3  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  -> 
( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
)
116, 7, 10syl2anc 397 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  ( <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  { <. x ,  y >.  |  ph } 
<->  ch ) )
125, 11mpbid 139 1  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   <.cop 3406   {copab 3845   Rel wrel 4378   ` cfv 4930   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by: (None)
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