Theorem resdifdir

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

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

Proof

Step	Hyp	Ref	Expression
1		indifdir	|- ( ( A \ B ) i^i ( C X. _V ) ) = ( ( A i^i ( C X. _V ) ) \ ( B i^i ( C X. _V ) ) )
2		df-res	|- ( ( A \ B ) |` C ) = ( ( A \ 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	difeq12i	|- ( ( A |` C ) \ ( B |` C ) ) = ( ( A i^i ( C X. _V ) ) \ ( B i^i ( C X. _V ) ) )
6	1 2 5	3eqtr4i	|- ( ( A \ B ) |` C ) = ( ( A |` C ) \ ( B |` C ) )