Theorem resundir 4833

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 3325	. 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 4551	. 2 |- ( ( A u. B ) |` C ) = ( ( A u. B ) i^i ( C X. _V ) )
3		df-res 4551	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
4		df-res 4551	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
5	3, 4	uneq12i 3228	. 2 |- ( ( A |` C ) u. ( B |` C ) ) = ( ( A i^i ( C X. _V ) ) u. ( B i^i ( C X. _V ) ) )
6	1, 2, 5	3eqtr4i 2170	1 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )

Syntax hints:   = wceq 1331   _Vcvv 2686   u. cun 3069   i^i cin 3070   X. cxp 4537   |` cres 4541

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-res 4551

This theorem is referenced by:  imaundir  4952  fvunsng  5614  fvsnun1  5617  fvsnun2  5618  fsnunfv  5621  fsnunres  5622  fseq1p1m1  9874  setsresg  11997  setscom  11999  setsslid  12009