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Theorem xpundi 4322
Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpundi  |-  ( A  X.  ( B  u.  C ) )  =  ( ( A  X.  B )  u.  ( A  X.  C ) )

Proof of Theorem xpundi
StepHypRef Expression
1 elun 2936 . . . . . 6  |-  ( y  e.  ( B  u.  C )  <->  ( y  e.  B  \/  y  e.  C ) )
21anbi2i 667 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
y  e.  B  \/  y  e.  C )
) )
3 andi 797 . . . . 5  |-  ( ( x  e.  A  /\  ( y  e.  B  \/  y  e.  C
) )  <->  ( (
x  e.  A  /\  y  e.  B )  \/  ( x  e.  A  /\  y  e.  C
) ) )
42, 3bitri 238 . . . 4  |-  ( ( x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( (
x  e.  A  /\  y  e.  B )  \/  ( x  e.  A  /\  y  e.  C
) ) )
54opabbii 3644 . . 3  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( B  u.  C ) ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  \/  ( x  e.  A  /\  y  e.  C ) ) }
6 unopab 3655 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  u.  {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) } )  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  \/  ( x  e.  A  /\  y  e.  C ) ) }
75, 6eqtr4i 2085 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( B  u.  C ) ) }  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  u.  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) } )
8 df-xp 4268 . 2  |-  ( A  X.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( B  u.  C
) ) }
9 df-xp 4268 . . 3  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
10 df-xp 4268 . . 3  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
119, 10uneq12i 2947 . 2  |-  ( ( A  X.  B )  u.  ( A  X.  C ) )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  u.  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) } )
127, 8, 113eqtr4i 2092 1  |-  ( A  X.  ( B  u.  C ) )  =  ( ( A  X.  B )  u.  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 355    /\ wa 356    = wceq 1518    e. wcel 1520    u. cun 2789   {copab 3637    X. cxp 4252
This theorem is referenced by:  xpun  4328  xp2cda  7265  xpcdaen  7268  alephadd  7667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1440  ax-6 1441  ax-7 1442  ax-gen 1443  ax-8 1522  ax-11 1523  ax-17 1527  ax-12o 1560  ax-10 1574  ax-9 1580  ax-4 1587  ax-16 1773  ax-ext 2044
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-ex 1445  df-sb 1734  df-clab 2050  df-cleq 2055  df-clel 2058  df-v 2475  df-un 2796  df-opab 3639  df-xp 4268
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