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Theorem xpopth 5830
Description: An ordered pair theorem for members of cross 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 5829 . . 3  |-  ( A  e.  ( C  X.  D )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2nd2 5829 . . 3  |-  ( B  e.  ( R  X.  S )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
31, 2eqeqan12d 2071 . 2  |-  ( ( A  e.  ( C  X.  D )  /\  B  e.  ( R  X.  S ) )  -> 
( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
4 1stexg 5822 . . . 4  |-  ( A  e.  ( C  X.  D )  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 5823 . . . 4  |-  ( A  e.  ( C  X.  D )  ->  ( 2nd `  A )  e. 
_V )
6 opthg 4003 . . . 4  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  -> 
( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
74, 5, 6syl2anc 397 . . 3  |-  ( A  e.  ( C  X.  D )  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
87adantr 265 . 2  |-  ( ( A  e.  ( C  X.  D )  /\  B  e.  ( R  X.  S ) )  -> 
( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
93, 8bitr2d 182 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 set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   <.cop 3406    X. cxp 4371   ` 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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