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Theorem iunin2 3720
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3710 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 2467 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
2 elin 3126 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32rexbii 2331 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  E. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 eliun 3661 . . . . 5  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
54anbi2i 430 . . . 4  |-  ( ( y  e.  B  /\  y  e.  U_ x  e.  A  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
61, 3, 53bitr4i 201 . . 3  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
7 eliun 3661 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  E. x  e.  A  y  e.  ( B  i^i  C ) )
8 elin 3126 . . 3  |-  ( y  e.  ( B  i^i  U_ x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
96, 7, 83bitr4i 201 . 2  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  y  e.  ( B  i^i  U_ x  e.  A  C )
)
109eqriv 2037 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 97    = wceq 1243    e. wcel 1393   E.wrex 2307    i^i cin 2916   U_ciun 3657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-iun 3659
This theorem is referenced by:  iunin1  3721  2iunin  3723  resiun1  4630  resiun2  4631
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