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

Theorem elopabi 6201
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 4828 . . . 4  |-  Rel  { <. x ,  y >.  |  ph }
2 1st2nd 6182 . . . 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 2371 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  ph } )
6 fvex 5555 . . 3  |-  ( 1st `  A )  e.  _V
7 fvex 5555 . . 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 4302 . 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 1632    e. wcel 1696   <.cop 3656   {copab 4092   Rel wrel 4710   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  drngoi  21090  vci  21120  com2i  25519  vri  25595  dicelval1sta  31999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139
  Copyright terms: Public domain W3C validator