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Theorem resundi 4968
Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
resundi  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )

Proof of Theorem resundi
StepHypRef Expression
1 xpundir 4741 . . . 4  |-  ( ( B  u.  C )  X.  _V )  =  ( ( B  X.  _V )  u.  ( C  X.  _V ) )
21ineq2i 3368 . . 3  |-  ( A  i^i  ( ( B  u.  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  u.  ( C  X.  _V ) ) )
3 indi 3416 . . 3  |-  ( A  i^i  ( ( B  X.  _V )  u.  ( C  X.  _V ) ) )  =  ( ( A  i^i  ( B  X.  _V )
)  u.  ( A  i^i  ( C  X.  _V ) ) )
42, 3eqtri 2304 . 2  |-  ( A  i^i  ( ( B  u.  C )  X. 
_V ) )  =  ( ( A  i^i  ( B  X.  _V )
)  u.  ( A  i^i  ( C  X.  _V ) ) )
5 df-res 4700 . 2  |-  ( A  |`  ( B  u.  C
) )  =  ( A  i^i  ( ( B  u.  C )  X.  _V ) )
6 df-res 4700 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
7 df-res 4700 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
86, 7uneq12i 3328 . 2  |-  ( ( A  |`  B )  u.  ( A  |`  C ) )  =  ( ( A  i^i  ( B  X.  _V ) )  u.  ( A  i^i  ( C  X.  _V )
) )
94, 5, 83eqtr4i 2314 1  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2789    u. cun 3151    i^i cin 3152    X. cxp 4686    |` cres 4690
This theorem is referenced by:  imaundi  5092  relresfld  5197  relcoi1  5199  resasplit  5377  fresaunres2  5379  fnsnsplit  5679  tfrlem16  6405  mapunen  7026  fnfi  7130  fseq1p1m1  10853  gsum2d  15219  dprd2da  15273  ptuncnv  17494  mbfres2  18996  evlseu  19396  cvmliftlem10  23232  eupath2lem3  23310  eldioph4b  26305  pwssplit4  26602
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 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
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 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-un 3158  df-in 3160  df-opab 4079  df-xp 4694  df-res 4700
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