Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is not empty) is a submanifold of $X$ with dimension $dimU+dimV-n$. Now if $X$ is a Whitney stratification and $U,V$ are Whitney substratifications, i.e. $U,V$ are subsets of $X$ with Whitney stratification structure, and each stratum of $U$ and $V$ contained in a single stratum of $X$. Assume that $U$ is transverse to $V$ in $X$, i.e. we require the usual transversal conditions on each stratum, then is it true that $U\cap V$ is a Whitney substratification of $X$ with dimension $dimU+dimV-n$? Where can I find some reference on this topic?

Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is not empty) is a submanifold of $X$ with dimension $\dim U+\dim V-n$. Now if $X$ is a Whitney stratification and $U,V$ are Whitney substratifications, i.e. $U$, $V$ are subsets of $X$ with Whitney stratification structure, and each stratum of $U$ and $V$ contained in a single stratum of $X$. Assume that $U$ is transverse to $V$ in $X$, i.e. we require the usual transversal conditions on each stratum, then is it true that $U\cap V$ is a Whitney substratification of $X$ with dimension $\dim U + \dim V - n$? Where can I find some reference on this topic?