measurable functions commute with conditional expectation

Question

Let $W_t = W (t)$ be a (one-dimensional) Wiener process, and fix an admissible filtration $\mathbb{F}.$ An adapted process $V_t$ is called elementary if it has the form $$V_t = \sum_{j = 0}^{k} \xi_j \chi_{(tj ,tj+1]}(t)$$ where $0 = t_0 < t_1 <···< t_k = 1,$ and for each index j the random variable $\xi_j$ is measurable relative to $\mathcal{F}_{t_j}.$ Define the Ito Integral of a simple process as follows: $$I_{t}(V) = \int_{0}^{t}V_{s}dW_{s} = \sum_{j=0}^{k}\xi_j(W_{t_{j+1}} - W_{t_{j}})$$ In some proofs of the properties of the Ito Integral, I see the following type of argument: (For simplicity, I will use a process with only one jump.) \begin{align*} \mathbb{E}\left(I_{t}(V)\right)&= \mathbb{E}(\xi_j(W_{t_{j+1}} - W_{t_{j}}))\\ &= \mathbb{E}\left[\mathbb{E}\left(\xi_j(W_{t_{j+1}} - W_{t_{j}})| \mathcal{F}_{t_{i+1}}\right)\right]\\ &=\mathbb{E}\xi_{j}\left[\mathbb{E}\left((W_{t_{j+1}} - W_{t_{j}})| \mathcal{F}_{t_{i+1}}\right)\right] \end{align*} I have been staring at this for a bit, but I am unsure where this string of equalities comes from. I am guessing that the second inequality comes from the Tower Property for conditional expectations? Also where the heck does the third equality come from? If we write this out we've \begin{align*} \int_{\Omega}I_{t}(V)d\mu &= \int_{\Omega}\xi_j(W_{t_{j+1}} - W_{t_{j}})d\mu\\ &= \int_{\Omega}\int_{\mathcal{F}_{t_{i+1}}}\xi_j(W_{t_{j+1}} - W_{t_{j}})d\mu d\mu \\ &=\int_{\Omega}\xi_{j}\int_{\mathcal{F}_{t_{i+1}}}(W_{t_{j+1}} - W_{t_{j}})d\mu d\mu \end{align*} where $(\Omega,\mu)$ is the background measure space. I still don't see why we can pull out the measurable function like that. Are my definitions wrong? Please help.