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Theorem resiun1 5111
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
resiun1  |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem resiun1
StepHypRef Expression
1 iunin2 4334 . 2  |-  U_ x  e.  A  ( ( C  X.  _V )  i^i 
B )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
2 df-res 4834 . . . . 5  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
3 incom 3631 . . . . 5  |-  ( B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  B )
42, 3eqtri 2431 . . . 4  |-  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B )
54a1i 11 . . 3  |-  ( x  e.  A  ->  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B ) )
65iuneq2i 4289 . 2  |-  U_ x  e.  A  ( B  |`  C )  =  U_ x  e.  A  (
( C  X.  _V )  i^i  B )
7 df-res 4834 . . 3  |-  ( U_ x  e.  A  B  |`  C )  =  (
U_ x  e.  A  B  i^i  ( C  X.  _V ) )
8 incom 3631 . . 3  |-  ( U_ x  e.  A  B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
97, 8eqtri 2431 . 2  |-  ( U_ x  e.  A  B  |`  C )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
101, 6, 93eqtr4ri 2442 1  |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   _Vcvv 3058    i^i cin 3412   U_ciun 4270    X. cxp 4820    |` cres 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-v 3060  df-in 3420  df-ss 3427  df-iun 4272  df-res 4834
This theorem is referenced by: (None)
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