  We prove a generalization of Thom's transversality theorem. It gives
conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M)\ra J^r(D,N)$ is
generically (for $f:M\ra N$) transverse to a submanifold $Z\subseteq J^r(D,N)$.
We apply this to study transversality properties of a restriction of a fixed
map $g:M\ra P$ to the preimage $(j^sf)^{-1}(A)$ of a submanifold $A\subseteq
J^s(M,N)$ in terms of transversality properties of the original map $f$. Our
main result is that for a reasonable class of submanifolds $A$ and a generic
map $f$ the restriction $g|_{(j^sf)^{-1}(A)}$ is also generic. We also present
an example of $A$ where the theorem fails.
