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Theorem xp2 6152
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 6149 . . 3  |-  ( x  e.  ( A  X.  B )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  B ) ) )
21abbi2i 2285 . 2  |-  ( A  X.  B )  =  { x  |  ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  B ) ) }
3 df-rab 2457 . 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 2194 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 1348    e. wcel 2141   {cab 2156   {crab 2452   _Vcvv 2730    X. cxp 4609   ` cfv 5198   1stc1st 6117   2ndc2nd 6118
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119  df-2nd 6120
This theorem is referenced by:  unielxp  6153
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