# Reference for truthful Nash on cartesian domain implies strategy-proofness

Consider a mechanism $M: \mathcal{R} \rightarrow X$, where

• $\mathcal{R}$ is a domain of preference profiles $R = (R_1,\dots, R_n)$, and
• $X$ is a set of outcomes.

I believe that the following is a folk theorem

If the domain $\mathcal{R}$ is a cartesian domain, i.e., $\mathcal{R} = \times_{i\in \{1,\dots,n\}} \mathcal{R}_i$, for some collection of individual domains $(\mathcal{R}_1,\dots, \mathcal{R}_n)$ and the game $(M,R)$ has a truthful Nash equilibrium for every profile $R \in \mathcal{R}$, then for every profile $R\in \mathcal{R}$, every player has a truthful dominant strategy in the game $(M,R)$.

This is not hard to prove and follows almost directly by definition, but I'd rather include a reference than a proof. I think I remember seeing this result proven somewhere (in a paper by Morris et al.?), but I cannot find that paper anymore.

Do you know any reference of a paper where that result (or a similar result) would be proven?