Theorem resundir 4905

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 3376	. 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 4623	. 2 |- ( ( A u. B ) |` C ) = ( ( A u. B ) i^i ( C X. _V ) )
3		df-res 4623	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
4		df-res 4623	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
5	3, 4	uneq12i 3279	. 2 |- ( ( A |` C ) u. ( B |` C ) ) = ( ( A i^i ( C X. _V ) ) u. ( B i^i ( C X. _V ) ) )
6	1, 2, 5	3eqtr4i 2201	1 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )

Syntax hints:   = wceq 1348   _Vcvv 2730   u. cun 3119   i^i cin 3120   X. cxp 4609   |` cres 4613

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-res 4623

This theorem is referenced by:  imaundir  5024  fvunsng  5690  fvsnun1  5693  fvsnun2  5694  fsnunfv  5697  fsnunres  5698  fseq1p1m1  10050  setsresg  12454  setscom  12456  setsslid  12466