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Theorem iinuni 3988
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem iinuni
StepHypRef Expression
1 r19.32v 2689 . . . 4  |-  ( A. x  e.  B  (
y  e.  A  \/  y  e.  x )  <->  ( y  e.  A  \/  A. x  e.  B  y  e.  x ) )
2 elun 3319 . . . . 5  |-  ( y  e.  ( A  u.  x )  <->  ( y  e.  A  \/  y  e.  x ) )
32ralbii 2570 . . . 4  |-  ( A. x  e.  B  y  e.  ( A  u.  x
)  <->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
4 vex 2794 . . . . . 6  |-  y  e. 
_V
54elint2 3872 . . . . 5  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
65orbi2i 507 . . . 4  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <-> 
( y  e.  A  \/  A. x  e.  B  y  e.  x )
)
71, 3, 63bitr4ri 271 . . 3  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <->  A. x  e.  B  y  e.  ( A  u.  x ) )
87abbii 2398 . 2  |-  { y  |  ( y  e.  A  \/  y  e. 
|^| B ) }  =  { y  | 
A. x  e.  B  y  e.  ( A  u.  x ) }
9 df-un 3160 . 2  |-  ( A  u.  |^| B )  =  { y  |  ( y  e.  A  \/  y  e.  |^| B ) }
10 df-iin 3911 . 2  |-  |^|_ x  e.  B  ( A  u.  x )  =  {
y  |  A. x  e.  B  y  e.  ( A  u.  x
) }
118, 9, 103eqtr4i 2316 1  |-  ( A  u.  |^| B )  = 
|^|_ x  e.  B  ( A  u.  x
)
Colors of variables: wff set class
Syntax hints:    \/ wo 359    = wceq 1625    e. wcel 1687   {cab 2272   A.wral 2546    u. cun 3153   |^|cint 3865   |^|_ciin 3909
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ral 2551  df-v 2793  df-un 3160  df-int 3866  df-iin 3911
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