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Theorem resundir 4916
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 3384 . 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 4634 . 2  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  u.  B
)  i^i  ( C  X.  _V ) )
3 df-res 4634 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
4 df-res 4634 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
53, 4uneq12i 3287 . 2  |-  ( ( A  |`  C )  u.  ( B  |`  C ) )  =  ( ( A  i^i  ( C  X.  _V ) )  u.  ( B  i^i  ( C  X.  _V )
) )
61, 2, 53eqtr4i 2208 1  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1353   _Vcvv 2737    u. cun 3127    i^i cin 3128    X. cxp 4620    |` cres 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-res 4634
This theorem is referenced by:  imaundir  5037  fvunsng  5705  fvsnun1  5708  fvsnun2  5709  fsnunfv  5712  fsnunres  5713  fseq1p1m1  10067  setsresg  12470  setscom  12472  setsslid  12482
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