Interpreting the edges in the Serre spectral sequence

Question

Let $F \hookrightarrow E \stackrel{\pi}{\rightarrow} B$ be a fiber bundle. For simplicity, assume that $F$ is connected, that $B$ is $1$-connected, and that $B$ is a CW complex. Consider the Serre spectral sequence $$E^2_{pq} = H_p(B;H_q(F)) \Rightarrow H_{p+q}(E).$$ By our assumptions, we have that $$E^2_{p0} = H_p(B) \quad \text{and} \quad E^2_{0q} = H_q(F).$$ I'd like to understand the terms $E^r_{p0}$ and $E^r_{0q}$ for $r \geq 2$. Each $E^{r+1}_{p0}$ is a subgroup of $E^r_{p0}$ and each $E^{r+1}_{0q}$ is a quotient of $E^r_{0q}$, so what we have are $$H_p(B) = E^2_{p0} \supset E^3_{p0} \supset E^4_{p0} \supset \cdots \supset E^{p+1}_{p0} = E^{\infty}_{p0}$$ and $$H_q(F) = E^2_{0q} \twoheadrightarrow E^3_{0q} \twoheadrightarrow E^4_{0q} \twoheadrightarrow \cdots \twoheadrightarrow E^{q+2}_{0q} = E^{\infty}_{0q}.$$ Now, one thing I know is that $$E^{\infty}_{p0} = E^{p+1}_{p0} = \text{Image}(H_p(E) \rightarrow H_p(B))$$ and that $$E^{\infty}_{0q} = E^{q+2}_{0q} = \text{Image}(H_q(F) \rightarrow H_q(E)).$$ Based on these known terms and the construction of the spectral sequence, my guess is that $E^r_{p0}$ and $E^r_{0q}$ are as follows. Let $B^{(k)}$ denote the $k$-skeleton of $B$ and let $E_k = \pi^{-1}(B^{(k)})$.

• I suspect that $E^r_{p0}$ is the image of $H_p(E,E_{p-r})$ in $H_p(B,B^{(p-r)})$, which for $r \geq 2$ is the same as $H_p(B)$. For $r = p+1$, we have $E_{p-r} = \emptyset$, so as we want this says that $E^{\infty}_{p0} = E^{p+1}_{p0}$ is the image of $H_p(E)$ in $H_p(B)$.
• I suspect that $E^r_{0q}$ is the image of $H_q(E_0)$ in $H_q(E_{r-1})$. To make sense of this, note that $E_0$ is the disjoint union of copies of $F$ indexed over the $0$-cells of $B$, and that the image of $H_q(E_0)$ in $H_q(E_1)$ is exactly $H_q(F)$. This uses the fact that $B$ is $1$-connected, so the action of $\pi_1(B)$ on $H_q(F)$ is trivial. For $r=q+2$, we have that $H_q(E_{r-1}) = H_q(E)$, so as we want this says that $E^{\infty}_{0q} = E^{q+2}_{0q}$ is the image of $H_q(F)$ in $H_q(E)$.

Question: Are these guesses correct, and if so what is a good reference for them?