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Theorem resiun1 4630
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 3720 . 2  |-  U_ x  e.  A  ( ( C  X.  _V )  i^i 
B )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
2 df-res 4357 . . . . 5  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
3 incom 3129 . . . . 5  |-  ( B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  B )
42, 3eqtri 2060 . . . 4  |-  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B )
54a1i 9 . . 3  |-  ( x  e.  A  ->  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B ) )
65iuneq2i 3675 . 2  |-  U_ x  e.  A  ( B  |`  C )  =  U_ x  e.  A  (
( C  X.  _V )  i^i  B )
7 df-res 4357 . . 3  |-  ( U_ x  e.  A  B  |`  C )  =  (
U_ x  e.  A  B  i^i  ( C  X.  _V ) )
8 incom 3129 . . 3  |-  ( U_ x  e.  A  B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
97, 8eqtri 2060 . 2  |-  ( U_ x  e.  A  B  |`  C )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
101, 6, 93eqtr4ri 2071 1  |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393   _Vcvv 2557    i^i cin 2916   U_ciun 3657    X. cxp 4343    |` cres 4347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659  df-res 4357
This theorem is referenced by: (None)
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