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Theorem xpopth 6823
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 6821 . . 3  |-  ( A  e.  ( C  X.  D )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2nd2 6821 . . 3  |-  ( B  e.  ( R  X.  S )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
31, 2eqeqan12d 2425 . 2  |-  ( ( A  e.  ( C  X.  D )  /\  B  e.  ( R  X.  S ) )  -> 
( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
4 fvex 5859 . . 3  |-  ( 1st `  A )  e.  _V
5 fvex 5859 . . 3  |-  ( 2nd `  A )  e.  _V
64, 5opth 4665 . 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 367    = wceq 1405    e. wcel 1842   <.cop 3978    X. cxp 4821   ` cfv 5569   1stc1st 6782   2ndc2nd 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fv 5577  df-1st 6784  df-2nd 6785
This theorem is referenced by:  fseqdom  8439  iundom2g  8947  mdetunilem9  19414  txhaus  20440  fsumvma  23869  2wlkeq  25124  disjxpin  27880  rmxypairf1o  35208
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