Theorem resundir 4898

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 3371	. 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 4616	. 2 |- ( ( A u. B ) |` C ) = ( ( A u. B ) i^i ( C X. _V ) )
3		df-res 4616	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
4		df-res 4616	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
5	3, 4	uneq12i 3274	. 2 |- ( ( A |` C ) u. ( B |` C ) ) = ( ( A i^i ( C X. _V ) ) u. ( B i^i ( C X. _V ) ) )
6	1, 2, 5	3eqtr4i 2196	1 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )

Syntax hints:   = wceq 1343   _Vcvv 2726   u. cun 3114   i^i cin 3115   X. cxp 4602   |` cres 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-res 4616

This theorem is referenced by:  imaundir  5017  fvunsng  5679  fvsnun1  5682  fvsnun2  5683  fsnunfv  5686  fsnunres  5687  fseq1p1m1  10029  setsresg  12432  setscom  12434  setsslid  12444