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Theorem resiun1 4974
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 3966 . 2  |-  U_ x  e.  A  ( ( C  X.  _V )  i^i 
B )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
2 df-res 4701 . . . . 5  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
3 incom 3361 . . . . 5  |-  ( B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  B )
42, 3eqtri 2303 . . . 4  |-  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B )
54a1i 10 . . 3  |-  ( x  e.  A  ->  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B ) )
65iuneq2i 3923 . 2  |-  U_ x  e.  A  ( B  |`  C )  =  U_ x  e.  A  (
( C  X.  _V )  i^i  B )
7 df-res 4701 . . 3  |-  ( U_ x  e.  A  B  |`  C )  =  (
U_ x  e.  A  B  i^i  ( C  X.  _V ) )
8 incom 3361 . . 3  |-  ( U_ x  e.  A  B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
97, 8eqtri 2303 . 2  |-  ( U_ x  e.  A  B  |`  C )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
101, 6, 93eqtr4ri 2314 1  |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   U_ciun 3905    X. cxp 4687    |` cres 4691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iun 3907  df-res 4701
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