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Theorem xpopth 6042
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 6041 . . 3  |-  ( A  e.  ( C  X.  D )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2nd2 6041 . . 3  |-  ( B  e.  ( R  X.  S )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
31, 2eqeqan12d 2133 . 2  |-  ( ( A  e.  ( C  X.  D )  /\  B  e.  ( R  X.  S ) )  -> 
( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
4 1stexg 6033 . . . 4  |-  ( A  e.  ( C  X.  D )  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 6034 . . . 4  |-  ( A  e.  ( C  X.  D )  ->  ( 2nd `  A )  e. 
_V )
6 opthg 4130 . . . 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 1316    e. wcel 1465   _Vcvv 2660   <.cop 3500    X. cxp 4507   ` cfv 5093   1stc1st 6004   2ndc2nd 6005
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-1st 6006  df-2nd 6007
This theorem is referenced by:  xmetxp  12603
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