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Theorem coundi 5087
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 4043 . . 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 4015 . . . . . . . 8  |-  ( x ( B  u.  C
) z  <->  ( x B z  \/  x C z ) )
32anbi1i 454 . . . . . . 7  |-  ( ( x ( B  u.  C ) z  /\  z A y )  <->  ( (
x B z  \/  x C z )  /\  z A y ) )
4 andir 809 . . . . . . 7  |-  ( ( ( x B z  \/  x C z )  /\  z A y )  <->  ( (
x B z  /\  z A y )  \/  ( x C z  /\  z A y ) ) )
53, 4bitri 183 . . . . . 6  |-  ( ( x ( B  u.  C ) z  /\  z A y )  <->  ( (
x B z  /\  z A y )  \/  ( x C z  /\  z A y ) ) )
65exbii 1585 . . . . 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 1608 . . . . 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 184 . . . 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 4031 . . 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 2178 . 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 4595 . . 3  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
12 df-co 4595 . . 3  |-  ( A  o.  C )  =  { <. x ,  y
>.  |  E. z
( x C z  /\  z A y ) }
1311, 12uneq12i 3259 . 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 4595 . 2  |-  ( A  o.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  E. z
( x ( B  u.  C ) z  /\  z A y ) }
1510, 13, 143eqtr4ri 2189 1  |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 698    = wceq 1335   E.wex 1472    u. cun 3100   class class class wbr 3965   {copab 4024    o. ccom 4590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-br 3966  df-opab 4026  df-co 4595
This theorem is referenced by:  relcoi1  5117
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