If $j : V \rightarrow M$ is an elementary embedding, what can we learn in $M$ from $j''ORD$? That is, what is $M[j''ORD]$?
Is it $M[j''ORD]$ equal to all of $V$?
If not, do we get a model intermediate between $M$ and $V$? If $\kappa$ is the critical point of $j$, is $\kappa$ still a large cardinal in $M[j''ORD]$? I am thinking of $j$ arising from a measure on $\kappa$, but I'm also interested in the more general situation.
My thinking goes like this: If we have the image of all of $V$, we can reconstruct $V$ itself by taking the Mostowski collapse of $j''V$ (and $j$ is the inverse of the Mostowski collapse). In $M[j''ORD]$, let's consider the class $W$ of sets with rank in $j''ORD$. Does the Mostowski collapse of $W$ yield all of $V$, and if not, what's missing?
EDIT: formatting, clarification of question.