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Theorem dfxp3 6058
Description: Define the cross product of three classes. Compare df-xp 4513. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
dfxp3  |-  ( ( A  X.  B )  X.  C )  =  { <. <. x ,  y
>. ,  z >.  |  ( x  e.  A  /\  y  e.  B  /\  z  e.  C
) }
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z

Proof of Theorem dfxp3
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 biidd 171 . . 3  |-  ( u  =  <. x ,  y
>.  ->  ( z  e.  C  <->  z  e.  C
) )
21dfoprab4 6056 . 2  |-  { <. u ,  z >.  |  ( u  e.  ( A  X.  B )  /\  z  e.  C ) }  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  e.  C
) }
3 df-xp 4513 . 2  |-  ( ( A  X.  B )  X.  C )  =  { <. u ,  z
>.  |  ( u  e.  ( A  X.  B
)  /\  z  e.  C ) }
4 df-3an 947 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  <->  ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C ) )
54oprabbii 5792 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ( x  e.  A  /\  y  e.  B  /\  z  e.  C ) }  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  e.  C
) }
62, 3, 53eqtr4i 2146 1  |-  ( ( A  X.  B )  X.  C )  =  { <. <. x ,  y
>. ,  z >.  |  ( x  e.  A  /\  y  e.  B  /\  z  e.  C
) }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463   <.cop 3498   {copab 3956    X. cxp 4505   {coprab 5741
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099  df-oprab 5744  df-1st 6004  df-2nd 6005
This theorem is referenced by: (None)
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