Theorem xpres

Description: A restriction of a Cartesian product is a Cartesian product. (Contributed by Stefan O'Rear, 24-Jan-2015) Extract from proof of xpssres . (Revised by BJ, 31-Oct-2026)

	Ref	Expression
Assertion	xpres	$$⊢{\left(A\times B\right)\upharpoonright }_C=\left(A\cap C\right)\times B$$

Proof

Step	Hyp	Ref	Expression
1		df-res	$$⊢{\left(A\times B\right)\upharpoonright }_C=\left(A\times B\right)\cap \left(C\times \mathrm{V}\right)$$
2		inxp	$$⊢\left(A\times B\right)\cap \left(C\times \mathrm{V}\right)=\left(A\cap C\right)\times \left(B\cap \mathrm{V}\right)$$
3		inv1	$$⊢B\cap \mathrm{V}=B$$
4	3	xpeq2i	$$⊢\left(A\cap C\right)\times \left(B\cap \mathrm{V}\right)=\left(A\cap C\right)\times B$$
5	1 2 4	3eqtri	$$⊢{\left(A\times B\right)\upharpoonright }_C=\left(A\cap C\right)\times B$$