We were given an exercise, to prove that for every regular $\theta$, the class $\{x||tc(x)|<\theta\}$ satisfies ZF-C without the powerset axiom. We ran into some difficulties proving the replacement axiom.
So, assuming all elements of the range are in $H_\theta$, we have no problem. But what guarantees us that this is the case? Can we nor work with functions that send elements in $H_\theta$ out of range?
The axiom schema of replacement (e.g. as described here) only applies to those $\phi$ such that for all $x$ we can find a unique $y$ such that $\phi(x,y)$ holds (omitting parameters $w_i$ for simplicity). This $y$ must thus exist and be unique in the model $V$ we are considering (so in particular it must be in $V$ or $H_\theta$), or we don't even need to consider this $\phi$.