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Theorem iunin2 4097
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4086 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
iunin2  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iunin2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.42v 2806 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
2 elin 3474 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32rexbii 2675 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  E. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 eliun 4040 . . . . 5  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
54anbi2i 676 . . . 4  |-  ( ( y  e.  B  /\  y  e.  U_ x  e.  A  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
61, 3, 53bitr4i 269 . . 3  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
7 eliun 4040 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  E. x  e.  A  y  e.  ( B  i^i  C ) )
8 elin 3474 . . 3  |-  ( y  e.  ( B  i^i  U_ x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
96, 7, 83bitr4i 269 . 2  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  y  e.  ( B  i^i  U_ x  e.  A  C )
)
109eqriv 2385 1  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651    i^i cin 3263   U_ciun 4036
This theorem is referenced by:  iunin1  4098  2iunin  4101  resiun1  5106  resiun2  5107  kmlem11  7974  cmpsublem  17385  cmpsub  17386  kgentopon  17492  metnrmlem3  18763  ovoliunlem1  19266  voliunlem1  19312  voliunlem2  19313  uniioombllem2  19343  uniioombllem4  19346  volsup2  19365  itg1addlem5  19460  itg1climres  19474  cvmscld  24740  heiborlem3  26214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-v 2902  df-in 3271  df-iun 4038
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