Notation: Suppose $\mathbf \Phi: U \rightarrow \mathbb R^m$ is a smooth map, with $U\subseteq \mathbb R^n$ an open set and $n\geq m$. The Jacobian is denoted by $\mathbf J_{\mathbf \Phi}(\cdot)$. Let $$k=max\left\lbrace \textrm{Rank}\hspace{1pt}{\mathbf J_{\mathbf \Phi}(\mathbf p)}\ \vert\ \mathbf p \in U\right\rbrace$$ be the maximal rank attained by the Jacobian in the domain, so that $k\leq m$.
Background: As stated in most books (e.g., Intro to Smooth Manifolds by Lee), the regular level set or preimage theorem (and in fact, also Sard's theorem) talks about regular values, where a regular value $\mathbf q \in \mathbb R^m$ is characterized by the property that everywhere in its preimage under $\mathbf \Phi$, $\mathbf J_{\mathbf \Phi}(\cdot)$ has a constant row-rank equal to $m$, i.e., is surjective.
My claim is that these theorems are also applicable if $k<m$. Just take a point $\tilde {\mathbf p}\in U$ such that $\mathbf J_{\mathbf \Phi}(\tilde {\mathbf p})$ has rank $k$. Now look at $\mathbf J_{\mathbf \Phi}(\tilde {\mathbf p})$ and pick the $ k$ linearly independent rows. Define $\tilde {\mathbf \Phi}:\mathbb R^n \rightarrow \mathbb R^{k}$ which retains only these rows. Now $\tilde {\mathbf \Phi}$ has regular values such as $\tilde{\mathbf \Phi} (\tilde {\mathbf p})$ that we can talk about, even though $\mathbf \Phi$ didn't.
Concern: Is there a reason why I'm unable to find the above (somewhat more general) form of these theorems? I can think of a few reasons, including (i) they trivially follow from the textbook versions of the theorems, or (ii) I'm missing some subtleties and my argument is wrong.