MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunin1 Unicode version

Theorem iunin1 4120
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4108 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 4119 . 2  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
2 incom 3497 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 11 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iuneq2i 4075 . 2  |-  U_ x  e.  A  ( C  i^i  B )  =  U_ x  e.  A  ( B  i^i  C )
5 incom 3497 . 2  |-  ( U_ x  e.  A  C  i^i  B )  =  ( B  i^i  U_ x  e.  A  C )
61, 4, 53eqtr4i 2438 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 1649    e. wcel 1721    i^i cin 3283   U_ciun 4057
This theorem is referenced by:  2iunin  4123  tgrest  17181  metnrmlem3  18848  limciun  19738  measinblem  24531  sstotbnd2  26377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-v 2922  df-in 3291  df-ss 3298  df-iun 4059
  Copyright terms: Public domain W3C validator