I have the theorem that if $f:\mathcal{M}\rightarrow\mathcal{N}$ is smooth, and $y\in{f(\mathcal{M})}\subset\mathcal{N}$ is a regular value, then $f^{-1}(y)$ is a proper submanifold of dimension $m-n$ (where $m,n$ are the dimensions of $\mathcal{M},\mathcal{N}$ respectively).
An example to demonstrate this is the map $f:T_{a,b}\rightarrow\mathbb{R}$ from the torus (defined as standard in $\mathbb{R}^3$, $a>b>0$, the circle of radius $b$ the one being rotated etc.) to the real line given by $f(x_1,x_2,x_3)=x_3$. This is just the height function. Now, intuitively, I can see why $f^{-1}(t)$ is going to be a submanifold of $T_{a,b}$ for $t\in(-a-b,-a+b)\cup(-a+b,a-b)\cup(a-b,a+b)$, since these preimages are essentially going to be either one circle, or two disjoint circles, 'wrapping' around the torus. At the two end points $(-a,-b)$ and $(a+b)$ the preimage is just a point, and in the intermediate points $(-a+b)$ and $(a-b)$, the preimage will be two circles just touching - so in these cases it is clear the preimages aren't submanifolds.
However, in trying to apply the theorem above, I compute the Jacobian of my function $f$ to be $(0,0, 1)$, which has maximal rank 1 for all $(x_1,x_2,x_3)$, i.e. it is surjective independently of $(x_1,x_2,x_3)$. So isn't this theorem telling me $f^{-1}(t)$ is a submanifold for all $t\in[-a-b,a+b]$, which is clearly wrong from what I've explained above?