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Theorem dfxp3 5946
Description: Define the cross product of three classes. Compare df-xp 4434. (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 170 . . 3  |-  ( u  =  <. x ,  y
>.  ->  ( z  e.  C  <->  z  e.  C
) )
21dfoprab4 5944 . 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 4434 . 2  |-  ( ( A  X.  B )  X.  C )  =  { <. u ,  z
>.  |  ( u  e.  ( A  X.  B
)  /\  z  e.  C ) }
4 df-3an 926 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  <->  ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C ) )
54oprabbii 5686 . 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 2118 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 102    /\ w3a 924    = wceq 1289    e. wcel 1438   <.cop 3444   {copab 3890    X. cxp 4426   {coprab 5635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-oprab 5638  df-1st 5893  df-2nd 5894
This theorem is referenced by: (None)
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