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Theorem rabun2 3447
Description: Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
Assertion
Ref Expression
rabun2  |-  { x  e.  ( A  u.  B
)  |  ph }  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph } )

Proof of Theorem rabun2
StepHypRef Expression
1 df-rab 2552 . 2  |-  { x  e.  ( A  u.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  u.  B )  /\  ph ) }
2 df-rab 2552 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2552 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
42, 3uneq12i 3327 . . 3  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
5 elun 3316 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 676 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ph ) )
7 andir 838 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) )
86, 7bitri 240 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph )
) )
98abbii 2395 . . . 4  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
10 unab 3435 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
119, 10eqtr4i 2306 . . 3  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
124, 11eqtr4i 2306 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }
131, 12eqtr4i 2306 1  |-  { x  e.  ( A  u.  B
)  |  ph }  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph } )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547    u. cun 3150
This theorem is referenced by:  fnsuppres  5732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-un 3157
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