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Theorem iinuni 4174
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 2854 . . . 4  |-  ( A. x  e.  B  (
y  e.  A  \/  y  e.  x )  <->  ( y  e.  A  \/  A. x  e.  B  y  e.  x ) )
2 elun 3488 . . . . 5  |-  ( y  e.  ( A  u.  x )  <->  ( y  e.  A  \/  y  e.  x ) )
32ralbii 2729 . . . 4  |-  ( A. x  e.  B  y  e.  ( A  u.  x
)  <->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
4 vex 2959 . . . . . 6  |-  y  e. 
_V
54elint2 4057 . . . . 5  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
65orbi2i 506 . . . 4  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <-> 
( y  e.  A  \/  A. x  e.  B  y  e.  x )
)
71, 3, 63bitr4ri 270 . . 3  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <->  A. x  e.  B  y  e.  ( A  u.  x ) )
87abbii 2548 . 2  |-  { y  |  ( y  e.  A  \/  y  e. 
|^| B ) }  =  { y  | 
A. x  e.  B  y  e.  ( A  u.  x ) }
9 df-un 3325 . 2  |-  ( A  u.  |^| B )  =  { y  |  ( y  e.  A  \/  y  e.  |^| B ) }
10 df-iin 4096 . 2  |-  |^|_ x  e.  B  ( A  u.  x )  =  {
y  |  A. x  e.  B  y  e.  ( A  u.  x
) }
118, 9, 103eqtr4i 2466 1  |-  ( A  u.  |^| B )  = 
|^|_ x  e.  B  ( A  u.  x
)
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705    u. cun 3318   |^|cint 4050   |^|_ciin 4094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-un 3325  df-int 4051  df-iin 4096
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