Is Fredholm operator always a closed map? Let $f:E\rightarrow F$ be a Fredholm operator between Banach spaces, then should $f$ always be a closed map? If this is not the case, then is it true that $f$ always maps a closed linear subspace to a closed linear subspace?

My claim is that every bounded linear operator $F: X \to Y$ ($X,Y$ Banach spaces) with finite-dimensional cokernel has closed image (i.e. $\mathrm{dim}(\mathrm{coker}(F))< \infty \implies \mathrm{im}(F)\subset Y$ closed).
Proof of the claim:
Define $n:=\mathrm{dim}(\mathrm{coker}(F)).$
Let us define the map
\begin{align*}
F_1 : X \oplus \mathbb R^n &\to Y,\\\space (x,\lambda) &\mapsto Fx +\sum_{1 \leq i \le n}\lambda_iy_i
\end{align*}
where $(y_i)_{i=1,...,n}$ is a tuple of representatives of a fixed basis of $Y/\mathrm{im}(F)$. Then, one sees that $F_1$ is surjective. Hence its image is the whole space $Y$, in particular $\mathrm{im}(F_1)$ is closed. Then, by the closed image theorem one can get a equivalent description of $\mathrm{im}(F_1)$ (closed) by the inequality:
$$\inf_{(z, \mu) \in \mathrm{ker}F_1} \Vert (x, \lambda)+(z, \mu) \Vert_{X \oplus \mathbb R^n} \leq C\Vert F_1(x, \lambda)\Vert_Y \space (\forall (x, \lambda) \in X\oplus \mathbb R^n)$$
for some $C > 0$. By taking $\lambda =0$, one then obtains:
$$\inf_{z\in \mathrm{ker}F} \Vert(x,0)+(z,0)\Vert_{X \oplus \mathbb R^n}=\inf_{z\in \mathrm{ker}F}\Vert x+z\Vert_X \leq C\Vert F_1(x,0)\Vert_Y=C\Vert F(x)\Vert_Y, \ \forall x \in X$$
implying that
$$\inf_{z\in \mathrm{ker}F} \Vert x+z \Vert_X \leq C\Vert F(x)\Vert_Y, \   \forall x \in X .$$
The last inequality then tells us that $\mathrm{im}(F)\subset Y$ is closed via closed image theorem. $\blacksquare$
Note: The only assumption we made is: $X, Y$ are Banach (we need that for the closed image theorem) and $F$ is a bounded linear operator between them with finite cokernel. So the assertion does not only hold for Fredholm operators ($\mathrm{dim}(\mathrm{ker}F)< \infty$ is not required for the proof), but also for a greater family of bounded linear operators between Banach spaces.
