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Theorem dfxp3 6747
Description: Define the Cartesian product of three classes. Compare df-xp 4957. (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 237 . . 3  |-  ( u  =  <. x ,  y
>.  ->  ( z  e.  C  <->  z  e.  C
) )
21dfoprab4 6744 . 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 4957 . 2  |-  ( ( A  X.  B )  X.  C )  =  { <. u ,  z
>.  |  ( u  e.  ( A  X.  B
)  /\  z  e.  C ) }
4 df-3an 967 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  <->  ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C ) )
54oprabbii 6253 . 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 2493 1  |-  ( ( A  X.  B )  X.  C )  =  { <. <. x ,  y
>. ,  z >.  |  ( x  e.  A  /\  y  e.  B  /\  z  e.  C
) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   <.cop 3994   {copab 4460    X. cxp 4949   {coprab 6204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-oprab 6207  df-1st 6690  df-2nd 6691
This theorem is referenced by: (None)
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