Theorem resdifcom

Description: Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021)

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

Proof

Step	Hyp	Ref	Expression
1		indif1	|- ( ( A \ C ) i^i ( B X. _V ) ) = ( ( A i^i ( B X. _V ) ) \ C )
2		df-res	|- ( ( A \ C ) |` B ) = ( ( A \ C ) i^i ( B X. _V ) )
3		df-res	|- ( A |` B ) = ( A i^i ( B X. _V ) )
4	3	difeq1i	|- ( ( A |` B ) \ C ) = ( ( A i^i ( B X. _V ) ) \ C )
5	1 2 4	3eqtr4ri	|- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )