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Theorem xpundir 4990
Description: Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
xpundir  |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )

Proof of Theorem xpundir
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4944 . 2  |-  ( ( A  u.  B )  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
2 df-xp 4944 . . . 4  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
3 df-xp 4944 . . . 4  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
42, 3uneq12i 3606 . . 3  |-  ( ( A  X.  C )  u.  ( B  X.  C ) )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) }  u.  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } )
5 elun 3595 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 695 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C ) )
7 andir 863 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C )  <->  ( (
x  e.  A  /\  y  e.  C )  \/  ( x  e.  B  /\  y  e.  C
) ) )
86, 7bitri 249 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) )
98opabbii 4454 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  e.  C ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) }
10 unopab 4465 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) }  u.  {
<. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } )  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) }
119, 10eqtr4i 2483 . . 3  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  e.  C ) }  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) }  u.  {
<. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } )
124, 11eqtr4i 2483 . 2  |-  ( ( A  X.  C )  u.  ( B  X.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
131, 12eqtr4i 2483 1  |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3424   {copab 4447    X. cxp 4936
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3070  df-un 3431  df-opab 4449  df-xp 4944
This theorem is referenced by:  xpun  4994  resundi  5222  xpfi  7684  cdaassen  8452  hashxplem  12297  ustund  19912  cnmpt2pc  20616  pwssplit4  29580  xpprsng  30857
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