Theorem resres

Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008)

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

Proof

Step	Hyp	Ref	Expression
1		df-res	|- ( ( A |` B ) |` C ) = ( ( A |` B ) i^i ( C X. _V ) )
2		df-res	|- ( A |` B ) = ( A i^i ( B X. _V ) )
3	2	ineq1i	|- ( ( A |` B ) i^i ( C X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) )
4		xpindir	|- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
5	4	ineq2i	|- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
6		df-res	|- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
7		inass	|- ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
8	5 6 7	3eqtr4ri	|- ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A |` ( B i^i C ) )
9	1 3 8	3eqtri	|- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )