Inverse image of differential of the tangent space of a submanifold.

Question

Let $M, N$ be smooth manifolds, $A \subseteq M$ a submanifold and $f : N \rightarrow M$ a smooth map such that $f^{-1}(A) \subseteq N$ is a submanifold (e.g. f transverse to $A$). Is it true in general that: $$ \forall y \in f^{-1}(A): (T_yf)^{-1}(T_{f(y)}A) = T_yf^{-1}(A)? $$ This seems to hold in simple examples I've come up with, but I have no idea how to (dis)prove this in general.