If $\alpha$ is an inaccessible cardinal and $V_{\alpha}$ the corresponding von Neumann universe then $V_{\alpha}$ is supposed to be a model of ZFC. But the singleton $\{V_{\alpha}\}$ is not in $V_{\alpha}$ because it has rank greater than $\alpha$. There is a bijection $\{1\} \to \{V_{\alpha}\}$ that sends $1$ to $V_{\alpha}$ so replacement should say that $\{V_{\alpha}\}$ is a "set", i.e., $\{V_{\alpha}\} \in V_{\alpha}$. Thus I've reached a contradiction.
I'm very new to thinking about set theory and I'm sure I've done something wrong because I misunderstood some definition or something. If anyone could set me straight I would appreciate it.
That $V_\alpha$ satisfies replacement means that $V_\alpha$ thinks there is a set that is $\{F(x)\mid x\in A\}$ whenever $A\in V_\alpha$ and $F$ is a class function that can be expressed in $V_\alpha$.
The latter condition means that the logical formula that represents $F$ is to be interpreted with all quantifiers ranging over $V_\alpha$ itself, and all parameters being elements of $V_\alpha$.
You cannot write your bijection $\{1\mapsto V_\alpha\}$ as a class function within $V_\alpha$ -- there's no way to even speak about the property of "being $V_\alpha$" for the function value, with a formula that's interpreted within $V_\alpha$ itself.