Theorem resindir

Description: Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008)

	Ref	Expression
Assertion	resindir	|- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )

Proof

Step	Hyp	Ref	Expression
1		inindir	|- ( ( A i^i B ) i^i ( C X. _V ) ) = ( ( A i^i ( C X. _V ) ) i^i ( B i^i ( C X. _V ) ) )
2		df-res	|- ( ( A i^i B ) |` C ) = ( ( A i^i 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	ineq12i	|- ( ( A |` C ) i^i ( B |` C ) ) = ( ( A i^i ( C X. _V ) ) i^i ( B i^i ( C X. _V ) ) )
6	1 2 5	3eqtr4i	|- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )