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Theorem xp2 6202
Description: Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
xp2  |-  ( A  X.  B )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  B ) }
Distinct variable groups:    x, A    x, B

Proof of Theorem xp2
StepHypRef Expression
1 elxp7 6199 . . 3  |-  ( x  e.  ( A  X.  B )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  B ) ) )
21abbi2i 2304 . 2  |-  ( A  X.  B )  =  { x  |  ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  B ) ) }
3 df-rab 2477 . 2  |-  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x )  e.  B
) }  =  {
x  |  ( x  e.  ( _V  X.  _V )  /\  (
( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  B ) ) }
42, 3eqtr4i 2213 1  |-  ( A  X.  B )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  B ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364    e. wcel 2160   {cab 2175   {crab 2472   _Vcvv 2752    X. cxp 4645   ` cfv 5238   1stc1st 6167   2ndc2nd 6168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fo 5244  df-fv 5246  df-1st 6169  df-2nd 6170
This theorem is referenced by:  unielxp  6203
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