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Theorem xpopth 6074
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 6073 . . 3  |-  ( A  e.  ( C  X.  D )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2nd2 6073 . . 3  |-  ( B  e.  ( R  X.  S )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
31, 2eqeqan12d 2155 . 2  |-  ( ( A  e.  ( C  X.  D )  /\  B  e.  ( R  X.  S ) )  -> 
( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
4 1stexg 6065 . . . 4  |-  ( A  e.  ( C  X.  D )  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 6066 . . . 4  |-  ( A  e.  ( C  X.  D )  ->  ( 2nd `  A )  e. 
_V )
6 opthg 4160 . . . 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 408 . . 3  |-  ( A  e.  ( C  X.  D )  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
87adantr 274 . 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 188 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 103    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2686   <.cop 3530    X. cxp 4537   ` cfv 5123   1stc1st 6036   2ndc2nd 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-1st 6038  df-2nd 6039
This theorem is referenced by:  xmetxp  12676
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