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Theorem coundi 5190
Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
coundi  |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C )
)

Proof of Theorem coundi
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 4111 . . 3  |-  ( {
<. x ,  y >.  |  E. z ( x B z  /\  z A y ) }  u.  { <. x ,  y >.  |  E. z ( x C z  /\  z A y ) } )  =  { <. x ,  y >.  |  ( E. z ( x B z  /\  z A y )  \/ 
E. z ( x C z  /\  z A y ) ) }
2 brun 4085 . . . . . . . 8  |-  ( x ( B  u.  C
) z  <->  ( x B z  \/  x C z ) )
32anbi1i 676 . . . . . . 7  |-  ( ( x ( B  u.  C ) z  /\  z A y )  <->  ( (
x B z  \/  x C z )  /\  z A y ) )
4 andir 838 . . . . . . 7  |-  ( ( ( x B z  \/  x C z )  /\  z A y )  <->  ( (
x B z  /\  z A y )  \/  ( x C z  /\  z A y ) ) )
53, 4bitri 240 . . . . . 6  |-  ( ( x ( B  u.  C ) z  /\  z A y )  <->  ( (
x B z  /\  z A y )  \/  ( x C z  /\  z A y ) ) )
65exbii 1572 . . . . 5  |-  ( E. z ( x ( B  u.  C ) z  /\  z A y )  <->  E. z
( ( x B z  /\  z A y )  \/  (
x C z  /\  z A y ) ) )
7 19.43 1595 . . . . 5  |-  ( E. z ( ( x B z  /\  z A y )  \/  ( x C z  /\  z A y ) )  <->  ( E. z ( x B z  /\  z A y )  \/  E. z ( x C z  /\  z A y ) ) )
86, 7bitr2i 241 . . . 4  |-  ( ( E. z ( x B z  /\  z A y )  \/ 
E. z ( x C z  /\  z A y ) )  <->  E. z ( x ( B  u.  C ) z  /\  z A y ) )
98opabbii 4099 . . 3  |-  { <. x ,  y >.  |  ( E. z ( x B z  /\  z A y )  \/ 
E. z ( x C z  /\  z A y ) ) }  =  { <. x ,  y >.  |  E. z ( x ( B  u.  C ) z  /\  z A y ) }
101, 9eqtri 2316 . 2  |-  ( {
<. x ,  y >.  |  E. z ( x B z  /\  z A y ) }  u.  { <. x ,  y >.  |  E. z ( x C z  /\  z A y ) } )  =  { <. x ,  y >.  |  E. z ( x ( B  u.  C ) z  /\  z A y ) }
11 df-co 4714 . . 3  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
12 df-co 4714 . . 3  |-  ( A  o.  C )  =  { <. x ,  y
>.  |  E. z
( x C z  /\  z A y ) }
1311, 12uneq12i 3340 . 2  |-  ( ( A  o.  B )  u.  ( A  o.  C ) )  =  ( { <. x ,  y >.  |  E. z ( x B z  /\  z A y ) }  u.  {
<. x ,  y >.  |  E. z ( x C z  /\  z A y ) } )
14 df-co 4714 . 2  |-  ( A  o.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  E. z
( x ( B  u.  C ) z  /\  z A y ) }
1510, 13, 143eqtr4ri 2327 1  |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C )
)
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632    u. cun 3163   class class class wbr 4039   {copab 4092    o. ccom 4709
This theorem is referenced by:  relcoi1  5217  cvmliftlem10  23840  diophren  26999  mvdco  27491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-br 4040  df-opab 4094  df-co 4714
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