I'm reading Jech's Set Theory, Chapter 17. Large Cardinals. I'm trying to follow the proof of Lemma 17.3 which states: If $j$ is a nontrivial elementary embedding of the universe, then there exists a measurable cardinal. We define $D$ to be the collection of subsets of $\kappa$ such that $X\in D$ if and only if $\kappa\in j(X)$ and show it's a $\kappa$-complete nonprincipal ultrafilter.
In the last paragraph (p.288) it says,
We shall now show that $D$ is $\kappa$-complete. Let $\gamma<\kappa$ and let $\mathcal{X}=\langle X_\alpha:\alpha<\gamma\rangle$ be a sequence of subsets of $\kappa$ such that $\kappa\in j(X_\alpha)$ for each $\alpha<\gamma$. We shall show that $\bigcap_{\alpha<\gamma}X_\alpha\in D$. In $M$ (and thus in $V$), $j(\mathcal{X})$ is a sequence of length $j(\gamma)$ of subsets of $j(\kappa)$; for each $\alpha<\gamma,$ the $j(\alpha)$th term of $j(\mathcal{X})$ is $j(X_\alpha)$. Since $j(\alpha)=\alpha$ for all $\alpha<\gamma$ and $j(\gamma)=\gamma$, it follows that $j(\mathcal{X})=\langle j(X_\alpha):\alpha<\gamma\rangle$. Hence if $X=\cap_{\alpha<\gamma}X_\alpha$, we have $j(X)=\cap_{\alpha<\gamma}j(X_\alpha)$. Now it is clear that $\kappa\in j(X)$ and hence $X\in D$.
I don't understand the parts that I boldfaced. First of all, how can it be that $j(\mathcal{X})$ is a sequence of length $j(\gamma)$ (even though it turns out that $j(\gamma)=\gamma$ anyway)? $\mathcal{X}$ is a sequence of length $\gamma$ so its image should also be a sequence of length $\gamma$, not $j(\gamma)$?
For any element $(\alpha,X_\alpha)$ of $\mathcal{X}$, do we have $j(\alpha,X_\alpha)=(j(\alpha),j(X_\alpha))$? Why is that?
And is $j(\bigcap_{\alpha<\gamma}X_\alpha)=\bigcap_{\alpha<\gamma}j(X_\alpha)$ because $j$ is an elementary embedding?
Lastly, what does he mean when he says "In $M$ (and thus in $V$), $j(\mathcal{X})$ is a sequence of length $j(\gamma)$..."?
“$\mathcal X$ is a sequence of length $\gamma$” is a perfectly good sentence with parameters $\mathcal X$ and $\gamma$ that holds in V, so the fact that $j$ is an elementary embedding immediately means that “$j(\mathcal X)$ is a sequence of length $j(\gamma)$” holds in $M$. Same reasoning implies that if $X$ is the $\alpha$-th element of $\mathcal X$, then $j(X)$ is the $j(\alpha)$-th element of $j(\mathcal X)$ (in $M$). By “thus in $V$” he simply means the relevant absoluteness result holds, in other words that if something is a sequence of given length in a model, it is so in any extension.
As for the part about intersections, think about how to define the intersection and what that translates to under the elementary embedding. The result follows from what has come before as well as the fact that $j(\gamma)=\gamma.$