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Theorem iunin1 4334
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4322 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iunin1  |-  U_ x  e.  A  ( C  i^i  B )  =  (
U_ x  e.  A  C  i^i  B )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iunin1
StepHypRef Expression
1 iunin2 4333 . 2  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
2 incom 3616 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 11 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iuneq2i 4288 . 2  |-  U_ x  e.  A  ( C  i^i  B )  =  U_ x  e.  A  ( B  i^i  C )
5 incom 3616 . 2  |-  ( U_ x  e.  A  C  i^i  B )  =  ( B  i^i  U_ x  e.  A  C )
61, 4, 53eqtr4i 2503 1  |-  U_ x  e.  A  ( C  i^i  B )  =  (
U_ x  e.  A  C  i^i  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    e. wcel 1904    i^i cin 3389   U_ciun 4269
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-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-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-iun 4271
This theorem is referenced by:  2iunin  4337  tgrest  20252  metnrmlem3  21956  metnrmlem3OLD  21971  limciun  22928  uniin1  28242  disjunsn  28281  measinblem  29116  sstotbnd2  32170  sge0iunmptlemre  38371
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