Theorem resdifdi

Description: Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024)

	Ref	Expression
Assertion	resdifdi	|- ( A |` ( B \ C ) ) = ( ( A |` B ) \ ( A |` C ) )

Proof

Step	Hyp	Ref	Expression
1		df-res	|- ( A |` ( B \ C ) ) = ( A i^i ( ( B \ C ) X. _V ) )
2		difxp1	|- ( ( B \ C ) X. _V ) = ( ( B X. _V ) \ ( C X. _V ) )
3	2	ineq2i	|- ( A i^i ( ( B \ C ) X. _V ) ) = ( A i^i ( ( B X. _V ) \ ( C X. _V ) ) )
4		indifdi	|- ( A i^i ( ( B X. _V ) \ ( C X. _V ) ) ) = ( ( A i^i ( B X. _V ) ) \ ( A i^i ( C X. _V ) ) )
5	1 3 4	3eqtri	|- ( A |` ( B \ C ) ) = ( ( A i^i ( B X. _V ) ) \ ( A i^i ( C X. _V ) ) )
6		df-res	|- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res	|- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6 7	difeq12i	|- ( ( A |` B ) \ ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) \ ( A i^i ( C X. _V ) ) )
9	5 8	eqtr4i	|- ( A |` ( B \ C ) ) = ( ( A |` B ) \ ( A |` C ) )