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Theorem iunin2 4389
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4378 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 3016 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
2 elin 3687 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32rexbii 2965 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  E. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 eliun 4330 . . . . 5  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
54anbi2i 694 . . . 4  |-  ( ( y  e.  B  /\  y  e.  U_ x  e.  A  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
61, 3, 53bitr4i 277 . . 3  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
7 eliun 4330 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  E. x  e.  A  y  e.  ( B  i^i  C ) )
8 elin 3687 . . 3  |-  ( y  e.  ( B  i^i  U_ x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
96, 7, 83bitr4i 277 . 2  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  y  e.  ( B  i^i  U_ x  e.  A  C )
)
109eqriv 2463 1  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    i^i cin 3475   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-in 3483  df-iun 4327
This theorem is referenced by:  iunin1  4390  2iunin  4393  resiun1  5290  resiun2  5291  infssuni  7807  kmlem11  8536  cmpsublem  19665  cmpsub  19666  kgentopon  19774  metnrmlem3  21100  ovoliunlem1  21648  voliunlem1  21695  voliunlem2  21696  uniioombllem2  21727  uniioombllem4  21730  volsup2  21749  itg1addlem5  21842  itg1climres  21856  cvmscld  28358  cnambfre  29640  ftc1anclem6  29672  heiborlem3  29912
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