Suppose there are infinitely many solutions to $f(x)=y$ in $\Omega$. From this collection of infinite solutions, select a sequence $\{x_{n}\}$ of distinct points, noting that $x_{n}\in\Omega$ and $f(x_{n}) = y$ for all $n$.
Since $\Omega$ is a bounded subset of $\mathbb{R}^{n}$, $\text{cl}(\Omega)$ is compact by the
Heine-Borel theorem. According to the
Bolzano-Weierstrass theorem, $\text{cl}(\Omega)$ is sequentially compact. Hence, $\{x_{n}\}$ admits a convergent subsequence whose limit, say $x$, is in $\text{cl}(\Omega)$. For ease of notation, we denote the convergent subsequence by $\{x_{n}\}$, as the original sequence is no longer needed.
Since $Df$ has constant rank at $x$, the
Constant Rank theorem says that there is a diffeomorphism $G$ defined on a neighborhood, say $U$, of $x$ sending $x$ to the origin of $\mathbb{R}^{n}$, and a diffeomorphism $F$ defined on a neighborhood of $f(x) = y$ sending $y$ to the origin of $\mathbb{R}^{m}$ such that $$(F\circ f\circ G^{-1})(x^{1},\ldots, x^{n}) = (x^{1},\ldots, x^{n}, \underbrace{0,\ldots, 0}_{m-n})\qquad (*)$$ Since the sequence of points we chose from the outset are distinct, since $U$ is a neighborhood of $x$, and since $x_{n}\rightarrow x$, there exists $N$ such that $x_{N}\in U$ and $x_{N}\neq x$. Since $x_{N}\neq x$ and since the diffeomorphism $G$ maps $x$ to the origin, we know $G(x_{N})\neq (\underbrace{0,\ldots, 0}_{n})$. Let $(p^{1}, \ldots, p^{n})$ be such that $x_{N} = G^{-1}(p^{1},\ldots, p^{n})$ and note that $(p^{1},\ldots, p^{n})\neq (\underbrace{0,\ldots, 0}_{n})$, as the previous sentence implies.
According to $(*)$, $$(F\circ f\circ G^{-1})(p^{1},\ldots, p^{n}) = (p^{1},\ldots, p^{n}, \underbrace{0,\ldots, 0}_{m-n})\neq (\underbrace{0,\ldots, 0}_{m})\qquad(**),$$ where the $\neq$ follows because $(p^{1},\ldots, p^{n})\neq (\underbrace{0,\ldots, 0}_{n})$.
However, we now have a contradiction because we know $f(x_{N})=y$ and that $F(y) = 0$. Thus, $$(F\circ f\circ G^{-1})(p^{1},\ldots, p^{n}) = (F\circ f)(x_{N}) = F(y) = (\underbrace{0,\ldots, 0}_{m}),$$ contradicting $(**)$.