Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-projun Structured version   Unicode version

Theorem bj-projun 31130
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 31127 . . . . 5  |-  ( A Proj 
B )  =  {
x  |  { x }  e.  ( B " { A } ) }
21abeq2i 2531 . . . 4  |-  ( x  e.  ( A Proj  B
)  <->  { x }  e.  ( B " { A } ) )
3 df-bj-proj 31127 . . . . 5  |-  ( A Proj 
C )  =  {
x  |  { x }  e.  ( C " { A } ) }
43abeq2i 2531 . . . 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 3586 . . 3  |-  ( x  e.  ( ( A Proj 
B )  u.  ( A Proj  C ) )  <->  ( x  e.  ( A Proj  B )  \/  x  e.  ( A Proj  C ) ) )
7 df-bj-proj 31127 . . . . 5  |-  ( A Proj  ( B  u.  C
) )  =  {
x  |  { x }  e.  ( ( B  u.  C ) " { A } ) }
87abeq2i 2531 . . . 4  |-  ( x  e.  ( A Proj  ( B  u.  C )
)  <->  { x }  e.  ( ( B  u.  C ) " { A } ) )
9 imaundir 5239 . . . . 5  |-  ( ( B  u.  C )
" { A }
)  =  ( ( B " { A } )  u.  ( C " { A }
) )
109eleq2i 2482 . . . 4  |-  ( { x }  e.  ( ( B  u.  C
) " { A } )  <->  { x }  e.  ( ( B " { A }
)  u.  ( C
" { A }
) ) )
11 elun 3586 . . . 4  |-  ( { x }  e.  ( ( B " { A } )  u.  ( C " { A }
) )  <->  ( {
x }  e.  ( B " { A } )  \/  {
x }  e.  ( C " { A } ) ) )
128, 10, 113bitri 273 . . 3  |-  ( x  e.  ( A Proj  ( B  u.  C )
)  <->  ( { x }  e.  ( B " { A } )  \/  { x }  e.  ( C " { A } ) ) )
135, 6, 123bitr4ri 280 . 2  |-  ( x  e.  ( A Proj  ( B  u.  C )
)  <->  x  e.  (
( A Proj  B )  u.  ( A Proj  C ) ) )
1413eqriv 2400 1  |-  ( A Proj  ( B  u.  C
) )  =  ( ( A Proj  B )  u.  ( A Proj  C
) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1407    e. wcel 1844    u. cun 3414   {csn 3974   "cima 4828   Proj bj-cproj 31126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-cnv 4833  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-bj-proj 31127
This theorem is referenced by:  bj-pr1un  31139  bj-pr2un  31153
  Copyright terms: Public domain W3C validator