Theorem resdifdir

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

	Ref	Expression
Assertion	resdifdir	$$⊢{\left(A\setminus B\right)\upharpoonright }_C=\left({A\upharpoonright }_C\right)\setminus \left({B\upharpoonright }_C\right)$$

Proof

Step	Hyp	Ref	Expression
1		indifdir	$$⊢\left(A\setminus B\right)\cap \left(C\times \mathrm{V}\right)=\left(A\cap \left(C\times \mathrm{V}\right)\right)\setminus \left(B\cap \left(C\times \mathrm{V}\right)\right)$$
2		df-res	$$⊢{\left(A\setminus B\right)\upharpoonright }_C=\left(A\setminus B\right)\cap \left(C\times \mathrm{V}\right)$$
3		df-res	$$⊢{A\upharpoonright }_C=A\cap \left(C\times \mathrm{V}\right)$$
4		df-res	$$⊢{B\upharpoonright }_C=B\cap \left(C\times \mathrm{V}\right)$$
5	3 4	difeq12i	$$⊢\left({A\upharpoonright }_C\right)\setminus \left({B\upharpoonright }_C\right)=\left(A\cap \left(C\times \mathrm{V}\right)\right)\setminus \left(B\cap \left(C\times \mathrm{V}\right)\right)$$
6	1 2 5	3eqtr4i	$$⊢{\left(A\setminus B\right)\upharpoonright }_C=\left({A\upharpoonright }_C\right)\setminus \left({B\upharpoonright }_C\right)$$