Theorem resundir

Description: Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004)

	Ref	Expression
Assertion	resundir	|- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )

Proof

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