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Theorem dfxp3 6397
Description: Define the cross product of three classes. Compare df-xp 4875. (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 229 . . 3  |-  ( u  =  <. x ,  y
>.  ->  ( z  e.  C  <->  z  e.  C
) )
21dfoprab4 6395 . 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 4875 . 2  |-  ( ( A  X.  B )  X.  C )  =  { <. u ,  z
>.  |  ( u  e.  ( A  X.  B
)  /\  z  e.  C ) }
4 df-3an 938 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  <->  ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C ) )
54oprabbii 6120 . 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 2465 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3809   {copab 4257    X. cxp 4867   {coprab 6073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-iota 5409  df-fun 5447  df-fv 5453  df-oprab 6076  df-1st 6340  df-2nd 6341
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