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

Theorem iunin1 4395
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4383 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 4394 . 2  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
2 incom 3696 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 11 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iuneq2i 4349 . 2  |-  U_ x  e.  A  ( C  i^i  B )  =  U_ x  e.  A  ( B  i^i  C )
5 incom 3696 . 2  |-  ( U_ x  e.  A  C  i^i  B )  =  ( B  i^i  U_ x  e.  A  C )
61, 4, 53eqtr4i 2506 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 1379    e. wcel 1767    i^i cin 3480   U_ciun 4330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-v 3120  df-in 3488  df-ss 3495  df-iun 4332
This theorem is referenced by:  2iunin  4398  tgrest  19505  metnrmlem3  21210  limciun  22143  disjunsn  27244  measinblem  27984  sstotbnd2  30165
  Copyright terms: Public domain W3C validator