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Theorem bj-projun 33850
Description: The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-projun  |-  ( A Proj  ( B  u.  C
) )  =  ( ( A Proj  B )  u.  ( A Proj  C
) )

Proof of Theorem bj-projun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-bj-proj 33847 . . . . 5  |-  ( A Proj 
B )  =  {
x  |  { x }  e.  ( B " { A } ) }
21abeq2i 2594 . . . 4  |-  ( x  e.  ( A Proj  B
)  <->  { x }  e.  ( B " { A } ) )
3 df-bj-proj 33847 . . . . 5  |-  ( A Proj 
C )  =  {
x  |  { x }  e.  ( C " { A } ) }
43abeq2i 2594 . . . 4  |-  ( x  e.  ( A Proj  C
)  <->  { x }  e.  ( C " { A } ) )
52, 4orbi12i 521 . . 3  |-  ( ( x  e.  ( A Proj 
B )  \/  x  e.  ( A Proj  C ) )  <->  ( { x }  e.  ( B " { A } )  \/  { x }  e.  ( C " { A } ) ) )
6 elun 3645 . . 3  |-  ( x  e.  ( ( A Proj 
B )  u.  ( A Proj  C ) )  <->  ( x  e.  ( A Proj  B )  \/  x  e.  ( A Proj  C ) ) )
7 df-bj-proj 33847 . . . . 5  |-  ( A Proj  ( B  u.  C
) )  =  {
x  |  { x }  e.  ( ( B  u.  C ) " { A } ) }
87abeq2i 2594 . . . 4  |-  ( x  e.  ( A Proj  ( B  u.  C )
)  <->  { x }  e.  ( ( B  u.  C ) " { A } ) )
9 imaundir 5419 . . . . 5  |-  ( ( B  u.  C )
" { A }
)  =  ( ( B " { A } )  u.  ( C " { A }
) )
109eleq2i 2545 . . . 4  |-  ( { x }  e.  ( ( B  u.  C
) " { A } )  <->  { x }  e.  ( ( B " { A }
)  u.  ( C
" { A }
) ) )
11 elun 3645 . . . 4  |-  ( { x }  e.  ( ( B " { A } )  u.  ( C " { A }
) )  <->  ( {
x }  e.  ( B " { A } )  \/  {
x }  e.  ( C " { A } ) ) )
128, 10, 113bitri 271 . . 3  |-  ( x  e.  ( A Proj  ( B  u.  C )
)  <->  ( { x }  e.  ( B " { A } )  \/  { x }  e.  ( C " { A } ) ) )
135, 6, 123bitr4ri 278 . 2  |-  ( x  e.  ( A Proj  ( B  u.  C )
)  <->  x  e.  (
( A Proj  B )  u.  ( A Proj  C ) ) )
1413eqriv 2463 1  |-  ( A Proj  ( B  u.  C
) )  =  ( ( A Proj  B )  u.  ( A Proj  C
) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1379    e. wcel 1767    u. cun 3474   {csn 4027   "cima 5002   Proj bj-cproj 33846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-bj-proj 33847
This theorem is referenced by:  bj-pr1un  33859  bj-pr2un  33873
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