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Theorem resundi 4957
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 4730 . . . 4  |-  ( ( B  u.  C )  X.  _V )  =  ( ( B  X.  _V )  u.  ( C  X.  _V ) )
21ineq2i 3342 . . 3  |-  ( A  i^i  ( ( B  u.  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  u.  ( C  X.  _V ) ) )
3 indi 3390 . . 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 2278 . 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 4681 . 2  |-  ( A  |`  ( B  u.  C
) )  =  ( A  i^i  ( ( B  u.  C )  X.  _V ) )
6 df-res 4681 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
7 df-res 4681 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
86, 7uneq12i 3302 . 2  |-  ( ( A  |`  B )  u.  ( A  |`  C ) )  =  ( ( A  i^i  ( B  X.  _V ) )  u.  ( A  i^i  ( C  X.  _V )
) )
94, 5, 83eqtr4i 2288 1  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   _Vcvv 2763    u. cun 3125    i^i cin 3126    X. cxp 4659    |` cres 4663
This theorem is referenced by:  imaundi  5081  relresfld  5186  relcoi1  5188  resasplit  5349  fresaunres2  5351  fnsnsplit  5651  tfrlem16  6377  mapunen  6998  fnfi  7102  fseq1p1m1  10824  gsum2d  15186  dprd2da  15240  ptuncnv  17461  mbfres2  18963  evlseu  19363  cvmliftlem10  23198  eupath2lem3  23276  eldioph4b  26262  pwssplit4  26559
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-un 3132  df-in 3134  df-opab 4052  df-xp 4675  df-res 4681
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