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Theorem resiun1 5165
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 4155 . 2  |-  U_ x  e.  A  ( ( C  X.  _V )  i^i 
B )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
2 df-res 4890 . . . . 5  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
3 incom 3533 . . . . 5  |-  ( B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  B )
42, 3eqtri 2456 . . . 4  |-  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B )
54a1i 11 . . 3  |-  ( x  e.  A  ->  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B ) )
65iuneq2i 4111 . 2  |-  U_ x  e.  A  ( B  |`  C )  =  U_ x  e.  A  (
( C  X.  _V )  i^i  B )
7 df-res 4890 . . 3  |-  ( U_ x  e.  A  B  |`  C )  =  (
U_ x  e.  A  B  i^i  ( C  X.  _V ) )
8 incom 3533 . . 3  |-  ( U_ x  e.  A  B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
97, 8eqtri 2456 . 2  |-  ( U_ x  e.  A  B  |`  C )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
101, 6, 93eqtr4ri 2467 1  |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319   U_ciun 4093    X. cxp 4876    |` cres 4880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-iun 4095  df-res 4890
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