Theorem resundir 4644

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

Step	Hyp	Ref	Expression
1		indir 3213	. 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 4375	. 2 |- ( ( A u. B ) |` C ) = ( ( A u. B ) i^i ( C X. _V ) )
3		df-res 4375	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
4		df-res 4375	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
5	3, 4	uneq12i 3124	. 2 |- ( ( A |` C ) u. ( B |` C ) ) = ( ( A i^i ( C X. _V ) ) u. ( B i^i ( C X. _V ) ) )
6	1, 2, 5	3eqtr4i 2111	1 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )

Syntax hints:   = wceq 1284   _Vcvv 2601   u. cun 2971   i^i cin 2972   X. cxp 4361   |` cres 4365

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-res 4375

This theorem is referenced by:  imaundir  4757  fvunsng  5378  fvsnun1  5381  fvsnun2  5382  fsnunfv  5384  fsnunres  5385  fseq1p1m1  9111