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

Theorem xpopth 6634
Description: An ordered pair theorem for members of Cartesian products. (Contributed by NM, 20-Jun-2007.)
Assertion
Ref Expression
xpopth  |-  ( ( A  e.  ( C  X.  D )  /\  B  e.  ( R  X.  S ) )  -> 
( ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A
)  =  ( 2nd `  B ) )  <->  A  =  B ) )

Proof of Theorem xpopth
StepHypRef Expression
1 1st2nd2 6632 . . 3  |-  ( A  e.  ( C  X.  D )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2nd2 6632 . . 3  |-  ( B  e.  ( R  X.  S )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
31, 2eqeqan12d 2458 . 2  |-  ( ( A  e.  ( C  X.  D )  /\  B  e.  ( R  X.  S ) )  -> 
( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
4 fvex 5720 . . 3  |-  ( 1st `  A )  e.  _V
5 fvex 5720 . . 3  |-  ( 2nd `  A )  e.  _V
64, 5opth 4585 . 2  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) )
73, 6syl6rbb 262 1  |-  ( ( A  e.  ( C  X.  D )  /\  B  e.  ( R  X.  S ) )  -> 
( ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A
)  =  ( 2nd `  B ) )  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3902    X. cxp 4857   ` cfv 5437   1stc1st 6594   2ndc2nd 6595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-iota 5400  df-fun 5439  df-fv 5445  df-1st 6596  df-2nd 6597
This theorem is referenced by:  fseqdom  8215  iundom2g  8723  mdetunilem9  18445  txhaus  19239  fsumvma  22571  disjxpin  25949  rmxypairf1o  29275  2wlkeq  30311
  Copyright terms: Public domain W3C validator