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Theorem coundi 5008
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 3975 . . 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 3947 . . . . . . . 8  |-  ( x ( B  u.  C
) z  <->  ( x B z  \/  x C z ) )
32anbi1i 451 . . . . . . 7  |-  ( ( x ( B  u.  C ) z  /\  z A y )  <->  ( (
x B z  \/  x C z )  /\  z A y ) )
4 andir 791 . . . . . . 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 1567 . . . . 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 1590 . . . . 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 3963 . . 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 2136 . 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 4516 . . 3  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
12 df-co 4516 . . 3  |-  ( A  o.  C )  =  { <. x ,  y
>.  |  E. z
( x C z  /\  z A y ) }
1311, 12uneq12i 3196 . 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 4516 . 2  |-  ( A  o.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  E. z
( x ( B  u.  C ) z  /\  z A y ) }
1510, 13, 143eqtr4ri 2147 1  |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 680    = wceq 1314   E.wex 1451    u. cun 3037   class class class wbr 3897   {copab 3956    o. ccom 4511
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-br 3898  df-opab 3958  df-co 4516
This theorem is referenced by:  relcoi1  5038
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