Understanding the Proof of the Smale Sard Theorem

Question

I am trying to understand the proof of Theorem 1.3 from https://people.math.osu.edu/burghelea.1/MaterialDiffTopology/Smale.pdf

It is stated as follows:

Let $f:M\to V$ be a $C^q$ Fredholm map with $q>\max(\operatorname{index}f,0)$. Then the regular values of $f$ form a residual subset of $V$.

For context for the proof, $U$ is an open subset of a Banach space $E$, $x_0\in U$, $A=Df(x_0):E\to E'$ and $E'$ is another Banach space. I understand why it is sufficient to prove the result locally. These are the parts of the proof I don't understand:

1. Since $\ker(A)$ is finite dimensional, $E$ can be written in the form $E_1\times \ker(A)$ (I understand why $\ker(A)$ is finite dimensional because $f$ is Fredholm but why does that imply the existence of $E_1$?)
2. The first partial derivative $D_1f(p,q):E_1\to E$ maps $E_1$ injectively onto a closed subspace of $E$ for all $(p,q)$ sufficiently close to $x_0\doteq(p_0,q_0)$ (I understand why $D_1f(p,q)(E_1)$ is a subspace because $D_1f$ is linear and $E_1$ is a subspace. I know that $D_1(f,q)(E_1)$ is closed because the range of a Fredholm operator is closed and I believe that $D_1f$ should be a Fredholm operator since $Df$ is a Fredholm operator. Why does $D_1f(p,q)$ need to be injective in a neighborhood of $x_0$?)
3. Using the implicit function theorem, we can choose a product neighborhood $C_1\times C_2$ of $(p_0,q_0)$ in $E_1\times \ker(A)$ such that $C_2$ is compact and if $q\in C_2$, $f$ restricted to $C_1\times \{q\}$ is a (differentiable) homeomorphism onto its image. (The previous point shows that $D_1f$ is injective. If we can show that $D_1f$ is also surjective then $D_1f$ is invertible and we can apply the Implicit Function Theorem. The Implicit Function Theorem would imply that there exists a neighborhood $C_1\subset E_1$ of $p_0$ and a unique continuously differentiable function $g:C_1\to \ker(A)$ such that $f(p,g(p))=0$ and $Dg(p)=(D_1f)^{-1}(p,g(p))D_2f(p,g(p))$ for every $p\in C_1$. Why does this imply the existence of $C_2$?)
4. Why does item 3 complete the proof of the theorem (the proof ends after this sentence)? We need to show that the regular values of $f$ restricted to a small enough open set is a residual set.