MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elopabi Unicode version

Theorem elopabi 6185
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 4812 . . . 4  |-  Rel  { <. x ,  y >.  |  ph }
2 1st2nd 6166 . . . 4  |-  ( ( Rel  { <. x ,  y >.  |  ph }  /\  A  e.  { <. x ,  y >.  |  ph } )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
31, 2mpan 651 . . 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 2358 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph } )
6 fvex 5539 . . 3  |-  ( 1st `  A )  e.  _V
7 fvex 5539 . . 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 4286 . 2  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
115, 10sylib 188 1  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   <.cop 3643   {copab 4076   Rel wrel 4694   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  drngoi  21074  vci  21104  com2i  25416  vri  25492  dicelval1sta  31377
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123
  Copyright terms: Public domain W3C validator