Proper definition of set between two functions

Question

I have a parameter $\delta \in (0,1)$. The parameter is further restricted from below and above by the functions $\underline \delta : (0,\ln(4)] \to (0,1)$ and $\overline \delta : (0,\ln(4)] \to (0,1)$ respectively where \begin{align} &\underline \delta(r) = \sup\left\{0,1 - \frac{\ln(2)}{r}\right\},\\[2mm] &\overline \delta(r) = 1 + \frac{\ln(3) - \ln(2+e^r)}{r}. \end{align} I want to define the set $A$ that captures all point $(r,\delta)$ such that $r \in (0,\ln(4)]$ and $\delta \in [\underline \delta(r), \overline \delta(r)]$. Is the following definition correct? \begin{align} (r,\delta) \in (0,\ln(4)] \times \left\{\delta \in [\underline \delta(r), \overline \delta(r)] \mid r \in (0,\ln(4)] \right\} =: A. \end{align} I'm a bit confused, because $r$ and $\delta$ turn up at the left and right hand side of $\in$

Answer 1

A = { $(r,\delta) : r \in (0,\ln(4)], \delta \in [\underline \delta(r), \overline \delta(r)]$ }.