How to show two sets with different numbers of degrees of freedom are different?

Question

Suppose there are two smooth injective functions $$f:\mathbb R^n \rightarrow \mathbb R^p \text{ and } g:\mathbb R^m \rightarrow \mathbb R^p, $$ where $m\neq n$. The domains of $f$ and $g$ are Cartesian products of intervals: $$D(f) = \times_{k=1}^{n}(a_k,b_k) \text{ and }D(g) = \times_{k=1}^{n}(c_k,d_k),$$ where $a_k,b_k,c_k,d_k \in [-\infty,+\infty]$. Is it possible to prove that the images of $f$ and $g$ are different, i.e. $$I(f) \neq I(g)?$$