Theorem resdifcom

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

	Ref	Expression
Assertion	resdifcom	$$⊢\left({A\upharpoonright }_B\right)\setminus C={\left(A\setminus C\right)\upharpoonright }_B$$

Proof

Step	Hyp	Ref	Expression
1		indif1	$$⊢\left(A\setminus C\right)\cap \left(B\times \mathrm{V}\right)=\left(A\cap \left(B\times \mathrm{V}\right)\right)\setminus C$$
2		df-res	$$⊢{\left(A\setminus C\right)\upharpoonright }_B=\left(A\setminus C\right)\cap \left(B\times \mathrm{V}\right)$$
3		df-res	$$⊢{A\upharpoonright }_B=A\cap \left(B\times \mathrm{V}\right)$$
4	3	difeq1i	$$⊢\left({A\upharpoonright }_B\right)\setminus C=\left(A\cap \left(B\times \mathrm{V}\right)\right)\setminus C$$
5	1 2 4	3eqtr4ri	$$⊢\left({A\upharpoonright }_B\right)\setminus C={\left(A\setminus C\right)\upharpoonright }_B$$