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Theorem resundir 4986
Description: Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
resundir  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )

Proof of Theorem resundir
StepHypRef Expression
1 indir 3430 . 2  |-  ( ( A  u.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  ( C  X.  _V ) )  u.  ( B  i^i  ( C  X.  _V )
) )
2 df-res 4717 . 2  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  u.  B
)  i^i  ( C  X.  _V ) )
3 df-res 4717 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
4 df-res 4717 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
53, 4uneq12i 3340 . 2  |-  ( ( A  |`  C )  u.  ( B  |`  C ) )  =  ( ( A  i^i  ( C  X.  _V ) )  u.  ( B  i^i  ( C  X.  _V )
) )
61, 2, 53eqtr4i 2326 1  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    u. cun 3163    i^i cin 3164    X. cxp 4703    |` cres 4707
This theorem is referenced by:  imaundir  5110  fresaunres2  5429  fvunsn  5728  fvsnun1  5731  fvsnun2  5732  fsnunfv  5736  fsnunres  5737  domss2  7036  axdc3lem4  8095  fseq1p1m1  10873  hashgval  11356  hashinf  11358  setsres  13190  setscom  13192  setsid  13203  ex-res  20844  wfrlem14  24340  mapfzcons1  26897  diophrw  26941  eldioph2lem1  26942  eldioph2lem2  26943  diophin  26955  pwssplit1  27291  pwssplit4  27294  constr3pthlem1  28401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-res 4717
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