TITLE: What does $\upharpoonright$ in $G(F\upharpoonright\alpha)$ mean?
QUESTION [8 upvotes]: More formally, we can state the Transfinite Recursion Theorem as follows.  Given a class function $G\colon V\to V$, there exists a unique transfinite sequence $F\colon\mathrm{Ord}\to V$ (where $\mathrm{Ord}$ is the class of all ordinals) such that $F(\alpha) = G(F\upharpoonright\alpha)$ for all ordinals $\alpha$. (Wikipedia, transfinite induction)

First question is, what does $\upharpoonright$ mean? Also, what exactly is $F$ in this usage? $F$ seems to be some form of function, but it says its transfinite sequence...

REPLY [3 votes]: If you are looking for a more concise general definition for restriction, here you are:
Suppose $F:X\rightarrow Y$ and $A\subseteq X$ then 
$F\restriction A=_\text{df}\{\langle x,y\rangle\in F\vert x\in A\}$
$F\restriction A$ is the restriction of $F$ to $A$