Theorem resdifdi

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

	Ref	Expression
Assertion	resdifdi	$$⊢{A\upharpoonright }_{\left(B\setminus C\right)}=\left({A\upharpoonright }_B\right)\setminus \left({A\upharpoonright }_C\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-res	$$⊢{A\upharpoonright }_{\left(B\setminus C\right)}=A\cap \left(\left(B\setminus C\right)\times \mathrm{V}\right)$$
2		difxp1	$$⊢\left(B\setminus C\right)\times \mathrm{V}=\left(B\times \mathrm{V}\right)\setminus \left(C\times \mathrm{V}\right)$$
3	2	ineq2i	$$⊢A\cap \left(\left(B\setminus C\right)\times \mathrm{V}\right)=A\cap \left(\left(B\times \mathrm{V}\right)\setminus \left(C\times \mathrm{V}\right)\right)$$
4		indifdi	$$⊢A\cap \left(\left(B\times \mathrm{V}\right)\setminus \left(C\times \mathrm{V}\right)\right)=\left(A\cap \left(B\times \mathrm{V}\right)\right)\setminus \left(A\cap \left(C\times \mathrm{V}\right)\right)$$
5	1 3 4	3eqtri	$$⊢{A\upharpoonright }_{\left(B\setminus C\right)}=\left(A\cap \left(B\times \mathrm{V}\right)\right)\setminus \left(A\cap \left(C\times \mathrm{V}\right)\right)$$
6		df-res	$$⊢{A\upharpoonright }_B=A\cap \left(B\times \mathrm{V}\right)$$
7		df-res	$$⊢{A\upharpoonright }_C=A\cap \left(C\times \mathrm{V}\right)$$
8	6 7	difeq12i	$$⊢\left({A\upharpoonright }_B\right)\setminus \left({A\upharpoonright }_C\right)=\left(A\cap \left(B\times \mathrm{V}\right)\right)\setminus \left(A\cap \left(C\times \mathrm{V}\right)\right)$$
9	5 8	eqtr4i	$$⊢{A\upharpoonright }_{\left(B\setminus C\right)}=\left({A\upharpoonright }_B\right)\setminus \left({A\upharpoonright }_C\right)$$