In [FSIT] and [OIMT] it is claimed that there is a surjection from $P(\kappa)\cap J^{\vec E}\_{\nu(E\_\alpha)}\times[\nu(E\_\alpha)]^{<\omega}$ onto $\alpha$, and that this surjection lies in $J\_{\alpha+1}^{\vec E}$. It is stated as a rather trivial fact, but I'm having trouble with seeing how this map should look like.


Here $\nu(E\_\alpha)$ is the natural length of the $(\kappa,\alpha)$ pre-extender $E\_\alpha$ and $\vec E$ is a fine extender sequence, the definition of which can be found in [OIMT] at page 11.


Since $E\_\alpha$ is the trivial completion of $E\_\alpha\upharpoonright\nu(E\_\alpha)$ I can see that $\nu(E\_\alpha)$ somehow "carries enough information" to determine $\alpha$, but this vague analogy just doesn't give me anything concrete. Of course, if a given surjection is found and is shown to be definable over $J\_\alpha^{\vec E}$, then it's in $J\_{\alpha+1}^{\vec E}$.


Thanks in advance!




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*References:*


* [FSIT] "Fine structure and iteration trees" by Steel and Mitchell
* [OIMT] "Outline of inner model theory" in the handbook, by Steel (preprint at <https://math.berkeley.edu/~steel/papers/steel1.pdf>)



Showing that $\alpha$ isn't a cardinal in $J_{\alpha+1}^{\vec E}$ for a fine extender sequence $\vec E$
We know that $\alpha = (\nu^{+})^{Ult(J^{\vec{E}}\_{\alpha}, E\_{\alpha})}$ and that $i\_{E\_{\alpha}} (\kappa) > \nu$, where $i\_{E\_{\alpha}}$ denotes the ultrapower embedding. Thus working in $Ult(J^{E\_{\alpha}}\_{\alpha}, E\_{\alpha})$ any $\beta <\nu^{+}=\alpha$ can be represented in the ultrapower $Ult(J^{\vec{E}}\_{\alpha}, E\_{\alpha})$ using a function $f: [\kappa]^{|a|} \rightarrow \kappa$, for some $a \in [\nu]^{<\omega}$ and $f \in J^{E\_{\alpha}}\_{\alpha}$. Thus we have a surjection from $(P(\kappa) \cap J^{E\_{\alpha}}\_{\alpha}) \times [\nu]^{<\omega}$ onto $\alpha$, via just looking at the representatives of ordinals less than $\alpha$ in the ultrapower.But the ultrapower can be constructed in $J^{\vec{E}}\_{\alpha +1}$ as it has all the information. So $\alpha$ can not be a cardinal in $J^{\vec{E}}\_{\alpha +1}$.


