Theorem resundi

Description: Distributive law for restriction over union. Theorem 31 of Suppes p. 65. (Contributed by NM, 30-Sep-2002)

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

Proof

Step	Hyp	Ref	Expression
1		xpundir	$$⊢\left(B\cup C\right)\times \mathrm{V}=\left(B\times \mathrm{V}\right)\cup \left(C\times \mathrm{V}\right)$$
2	1	ineq2i	$$⊢A\cap \left(\left(B\cup C\right)\times \mathrm{V}\right)=A\cap \left(\left(B\times \mathrm{V}\right)\cup \left(C\times \mathrm{V}\right)\right)$$
3		indi	$$⊢A\cap \left(\left(B\times \mathrm{V}\right)\cup \left(C\times \mathrm{V}\right)\right)=\left(A\cap \left(B\times \mathrm{V}\right)\right)\cup \left(A\cap \left(C\times \mathrm{V}\right)\right)$$
4	2 3	eqtri	$$⊢A\cap \left(\left(B\cup C\right)\times \mathrm{V}\right)=\left(A\cap \left(B\times \mathrm{V}\right)\right)\cup \left(A\cap \left(C\times \mathrm{V}\right)\right)$$
5		df-res	$$⊢{A\upharpoonright }_{\left(B\cup C\right)}=A\cap \left(\left(B\cup C\right)\times \mathrm{V}\right)$$
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	uneq12i	$$⊢\left({A\upharpoonright }_B\right)\cup \left({A\upharpoonright }_C\right)=\left(A\cap \left(B\times \mathrm{V}\right)\right)\cup \left(A\cap \left(C\times \mathrm{V}\right)\right)$$
9	4 5 8	3eqtr4i	$$⊢{A\upharpoonright }_{\left(B\cup C\right)}=\left({A\upharpoonright }_B\right)\cup \left({A\upharpoonright }_C\right)$$