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Theorem iunin1 3969
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3957 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 3968 . 2  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
2 incom 3363 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 12 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iuneq2i 3925 . 2  |-  U_ x  e.  A  ( C  i^i  B )  =  U_ x  e.  A  ( B  i^i  C )
5 incom 3363 . 2  |-  ( U_ x  e.  A  C  i^i  B )  =  ( B  i^i  U_ x  e.  A  C )
61, 4, 53eqtr4i 2315 1  |-  U_ x  e.  A  ( C  i^i  B )  =  (
U_ x  e.  A  C  i^i  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1624    e. wcel 1685    i^i cin 3153   U_ciun 3907
This theorem is referenced by:  2iunin  3972  tgrest  16885  metnrmlem3  18360  limciun  19239  sstotbnd2  25898
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rex 2551  df-v 2792  df-in 3161  df-ss 3168  df-iun 3909
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