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 ) ∩ ( 𝐶 × V ) ) = ( ( 𝐴 ∩ ( 𝐶 × V ) ) ∖ ( 𝐵 ∩ ( 𝐶 × V ) ) )
2		df-res	⊢ ( ( 𝐴 ∖ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐶 × V ) )
3		df-res	⊢ ( 𝐴 ↾ 𝐶 ) = ( 𝐴 ∩ ( 𝐶 × V ) )
4		df-res	⊢ ( 𝐵 ↾ 𝐶 ) = ( 𝐵 ∩ ( 𝐶 × V ) )
5	3 4	difeq12i	⊢ ( ( 𝐴 ↾ 𝐶 ) ∖ ( 𝐵 ↾ 𝐶 ) ) = ( ( 𝐴 ∩ ( 𝐶 × V ) ) ∖ ( 𝐵 ∩ ( 𝐶 × V ) ) )
6	1 2 5	3eqtr4i	⊢ ( ( 𝐴 ∖ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ↾ 𝐶 ) ∖ ( 𝐵 ↾ 𝐶 ) )