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Theorem xpundir 4903
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 4855 . 2  |-  ( ( A  u.  B )  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
2 df-xp 4855 . . . 4  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
3 df-xp 4855 . . . 4  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
42, 3uneq12i 3618 . . 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 3606 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 699 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C ) )
7 andir 876 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C )  <->  ( (
x  e.  A  /\  y  e.  C )  \/  ( x  e.  B  /\  y  e.  C
) ) )
86, 7bitri 252 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) )
98opabbii 4485 . . . 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 4496 . . . 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 2454 . . 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 2454 . 2  |-  ( ( A  X.  C )  u.  ( B  X.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
131, 12eqtr4i 2454 1  |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1868    u. cun 3434   {copab 4478    X. cxp 4847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-un 3441  df-opab 4480  df-xp 4855
This theorem is referenced by:  xpun  4907  resundi  5133  xpfi  7844  cdaassen  8612  hashxplem  12602  ustund  21222  cnmpt2pc  21942  poimirlem3  31856  poimirlem4  31857  poimirlem6  31859  poimirlem7  31860  poimirlem16  31869  poimirlem19  31872  pwssplit4  35866  xpprsng  39386
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