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Theorem dfxp3 5848
Description: Define the cross product of three classes. Compare df-xp 4379. (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 165 . . 3  |-  ( u  =  <. x ,  y
>.  ->  ( z  e.  C  <->  z  e.  C
) )
21dfoprab4 5846 . 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 4379 . 2  |-  ( ( A  X.  B )  X.  C )  =  { <. u ,  z
>.  |  ( u  e.  ( A  X.  B
)  /\  z  e.  C ) }
4 df-3an 898 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  <->  ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C ) )
54oprabbii 5588 . 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 2086 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 101    /\ w3a 896    = wceq 1259    e. wcel 1409   <.cop 3406   {copab 3845    X. cxp 4371   {coprab 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-oprab 5544  df-1st 5795  df-2nd 5796
This theorem is referenced by: (None)
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