Theorem resres

Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008)

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

Proof

Step	Hyp	Ref	Expression
1		df-res	$$⊢{\left({A\upharpoonright }_B\right)\upharpoonright }_C=\left({A\upharpoonright }_B\right)\cap \left(C\times \mathrm{V}\right)$$
2		df-res	$$⊢{A\upharpoonright }_B=A\cap \left(B\times \mathrm{V}\right)$$
3	2	ineq1i	$$⊢\left({A\upharpoonright }_B\right)\cap \left(C\times \mathrm{V}\right)=\left(A\cap \left(B\times \mathrm{V}\right)\right)\cap \left(C\times \mathrm{V}\right)$$
4		xpindir	$$⊢\left(B\cap C\right)\times \mathrm{V}=\left(B\times \mathrm{V}\right)\cap \left(C\times \mathrm{V}\right)$$
5	4	ineq2i	$$⊢A\cap \left(\left(B\cap C\right)\times \mathrm{V}\right)=A\cap \left(\left(B\times \mathrm{V}\right)\cap \left(C\times \mathrm{V}\right)\right)$$
6		df-res	$$⊢{A\upharpoonright }_{\left(B\cap C\right)}=A\cap \left(\left(B\cap C\right)\times \mathrm{V}\right)$$
7		inass	$$⊢\left(A\cap \left(B\times \mathrm{V}\right)\right)\cap \left(C\times \mathrm{V}\right)=A\cap \left(\left(B\times \mathrm{V}\right)\cap \left(C\times \mathrm{V}\right)\right)$$
8	5 6 7	3eqtr4ri	$$⊢\left(A\cap \left(B\times \mathrm{V}\right)\right)\cap \left(C\times \mathrm{V}\right)={A\upharpoonright }_{\left(B\cap C\right)}$$
9	1 3 8	3eqtri	$$⊢{\left({A\upharpoonright }_B\right)\upharpoonright }_C={A\upharpoonright }_{\left(B\cap C\right)}$$