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Theorem iunin2 3778
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3768 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 2520 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
2 elin 3172 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32rexbii 2381 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  E. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 eliun 3719 . . . . 5  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
54anbi2i 445 . . . 4  |-  ( ( y  e.  B  /\  y  e.  U_ x  e.  A  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
61, 3, 53bitr4i 210 . . 3  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
7 eliun 3719 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  E. x  e.  A  y  e.  ( B  i^i  C ) )
8 elin 3172 . . 3  |-  ( y  e.  ( B  i^i  U_ x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
96, 7, 83bitr4i 210 . 2  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  y  e.  ( B  i^i  U_ x  e.  A  C )
)
109eqriv 2082 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 102    = wceq 1287    e. wcel 1436   E.wrex 2356    i^i cin 2987   U_ciun 3715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-in 2994  df-iun 3717
This theorem is referenced by:  iunin1  3779  2iunin  3781  resiun1  4701  resiun2  4702
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