Theorem xpdisjres

Description: Restriction of a constant function (or other Cartesian product) outside of its domain. (Contributed by Thierry Arnoux, 25-Jan-2017)

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

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		xpdisj1	$$⊢A\cap C=\varnothing \to \left(A\times B\right)\cap \left(C\times \mathrm{V}\right)=\varnothing$$
3	1 2	syl5eq	$$⊢A\cap C=\varnothing \to {\left(A\times B\right)\upharpoonright }_C=\varnothing$$