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Theorem iinuni 4358
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 2922 . . . 4  |-  ( A. x  e.  B  (
y  e.  A  \/  y  e.  x )  <->  ( y  e.  A  \/  A. x  e.  B  y  e.  x ) )
2 elun 3565 . . . . 5  |-  ( y  e.  ( A  u.  x )  <->  ( y  e.  A  \/  y  e.  x ) )
32ralbii 2823 . . . 4  |-  ( A. x  e.  B  y  e.  ( A  u.  x
)  <->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
4 vex 3034 . . . . . 6  |-  y  e. 
_V
54elint2 4233 . . . . 5  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
65orbi2i 528 . . . 4  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <-> 
( y  e.  A  \/  A. x  e.  B  y  e.  x )
)
71, 3, 63bitr4ri 286 . . 3  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <->  A. x  e.  B  y  e.  ( A  u.  x ) )
87abbii 2587 . 2  |-  { y  |  ( y  e.  A  \/  y  e. 
|^| B ) }  =  { y  | 
A. x  e.  B  y  e.  ( A  u.  x ) }
9 df-un 3395 . 2  |-  ( A  u.  |^| B )  =  { y  |  ( y  e.  A  \/  y  e.  |^| B ) }
10 df-iin 4272 . 2  |-  |^|_ x  e.  B  ( A  u.  x )  =  {
y  |  A. x  e.  B  y  e.  ( A  u.  x
) }
118, 9, 103eqtr4i 2503 1  |-  ( A  u.  |^| B )  = 
|^|_ x  e.  B  ( A  u.  x
)
Colors of variables: wff setvar class
Syntax hints:    \/ wo 375    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756    u. cun 3388   |^|cint 4226   |^|_ciin 4270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-un 3395  df-int 4227  df-iin 4272
This theorem is referenced by: (None)
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