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Theorem iinuni 3985
Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iinuni  |-  ( A  u.  |^| B )  = 
|^|_ x  e.  B  ( A  u.  x
)
Distinct variable groups:    x, A    x, B

Proof of Theorem iinuni
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.32v 2686 . . . 4  |-  ( A. x  e.  B  (
y  e.  A  \/  y  e.  x )  <->  ( y  e.  A  \/  A. x  e.  B  y  e.  x ) )
2 elun 3316 . . . . 5  |-  ( y  e.  ( A  u.  x )  <->  ( y  e.  A  \/  y  e.  x ) )
32ralbii 2567 . . . 4  |-  ( A. x  e.  B  y  e.  ( A  u.  x
)  <->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
4 vex 2791 . . . . . 6  |-  y  e. 
_V
54elint2 3869 . . . . 5  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
65orbi2i 505 . . . 4  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <-> 
( y  e.  A  \/  A. x  e.  B  y  e.  x )
)
71, 3, 63bitr4ri 269 . . 3  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <->  A. x  e.  B  y  e.  ( A  u.  x ) )
87abbii 2395 . 2  |-  { y  |  ( y  e.  A  \/  y  e. 
|^| B ) }  =  { y  | 
A. x  e.  B  y  e.  ( A  u.  x ) }
9 df-un 3157 . 2  |-  ( A  u.  |^| B )  =  { y  |  ( y  e.  A  \/  y  e.  |^| B ) }
10 df-iin 3908 . 2  |-  |^|_ x  e.  B  ( A  u.  x )  =  {
y  |  A. x  e.  B  y  e.  ( A  u.  x
) }
118, 9, 103eqtr4i 2313 1  |-  ( A  u.  |^| B )  = 
|^|_ x  e.  B  ( A  u.  x
)
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    u. cun 3150   |^|cint 3862   |^|_ciin 3906
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-ral 2548  df-v 2790  df-un 3157  df-int 3863  df-iin 3908
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