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Theorem xp2 6121
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 6118 . . 3  |-  ( x  e.  ( A  X.  B )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  B ) ) )
21abbi2i 2272 . 2  |-  ( A  X.  B )  =  { x  |  ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  B ) ) }
3 df-rab 2444 . 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 2181 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 103    = wceq 1335    e. wcel 2128   {cab 2143   {crab 2439   _Vcvv 2712    X. cxp 4584   ` cfv 5170   1stc1st 6086   2ndc2nd 6087
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fo 5176  df-fv 5178  df-1st 6088  df-2nd 6089
This theorem is referenced by:  unielxp  6122
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