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Theorem coundi 5112
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 4068 . . 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 4040 . . . . . . . 8  |-  ( x ( B  u.  C
) z  <->  ( x B z  \/  x C z ) )
32anbi1i 455 . . . . . . 7  |-  ( ( x ( B  u.  C ) z  /\  z A y )  <->  ( (
x B z  \/  x C z )  /\  z A y ) )
4 andir 814 . . . . . . 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 1598 . . . . 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 1621 . . . . 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 4056 . . 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 2191 . 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 4620 . . 3  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
12 df-co 4620 . . 3  |-  ( A  o.  C )  =  { <. x ,  y
>.  |  E. z
( x C z  /\  z A y ) }
1311, 12uneq12i 3279 . 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 4620 . 2  |-  ( A  o.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  E. z
( x ( B  u.  C ) z  /\  z A y ) }
1510, 13, 143eqtr4ri 2202 1  |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 703    = wceq 1348   E.wex 1485    u. cun 3119   class class class wbr 3989   {copab 4049    o. ccom 4615
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-br 3990  df-opab 4051  df-co 4620
This theorem is referenced by:  relcoi1  5142
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