MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2iunin Structured version   Unicode version

Theorem 2iunin 4364
Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
2iunin  |-  U_ x  e.  A  U_ y  e.  B  ( C  i^i  D )  =  ( U_ x  e.  A  C  i^i  U_ y  e.  B  D )
Distinct variable groups:    x, B    y, C    x, D    x, y
Allowed substitution hints:    A( x, y)    B( y)    C( x)    D( y)

Proof of Theorem 2iunin
StepHypRef Expression
1 iunin2 4360 . . . 4  |-  U_ y  e.  B  ( C  i^i  D )  =  ( C  i^i  U_ y  e.  B  D )
21a1i 11 . . 3  |-  ( x  e.  A  ->  U_ y  e.  B  ( C  i^i  D )  =  ( C  i^i  U_ y  e.  B  D )
)
32iuneq2i 4315 . 2  |-  U_ x  e.  A  U_ y  e.  B  ( C  i^i  D )  =  U_ x  e.  A  ( C  i^i  U_ y  e.  B  D )
4 iunin1 4361 . 2  |-  U_ x  e.  A  ( C  i^i  U_ y  e.  B  D )  =  (
U_ x  e.  A  C  i^i  U_ y  e.  B  D )
53, 4eqtri 2451 1  |-  U_ x  e.  A  U_ y  e.  B  ( C  i^i  D )  =  ( U_ x  e.  A  C  i^i  U_ y  e.  B  D )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1868    i^i cin 3435   U_ciun 4296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-v 3083  df-in 3443  df-ss 3450  df-iun 4298
This theorem is referenced by:  fpar  6908
  Copyright terms: Public domain W3C validator