This questions concerns notation. I have a $2\times n$ matrix $X$. Is there a smooth way to denote the minimum of the $2$nd row conditional on elements in the first row being equal to some constant $c$. For example if $$X = \left(\begin{array}{ccc} 0 & 1 & 0 & 1 \\ 0.7 & 0.4 & 0.5 & 0.1\end{array}\right)$$ then for $c = 0$ is the demanded minimum $0.5$ (and $0.1$ for $c = 1$).
Background: I am parsing sequentially through a list. Every list entry that satisfies $k$ conditions (in the example there is only 1 condition, the first row), I want to delete those entries for which another variable (in the example the second row) is minimal.
Simply spelling out the set over which you want to take the minimum seems reasonable. For example, write your matrix $X$ as $X = (x_{ij})_{2\times n}$, where $x_{ij}$ is the entry in the $i$-th row and $j$-th column of $X$. Then, for a fixed value of $c$, you are trying to determine the minimal element of the second row such that the entry just above that is $c$. In notation, something like $$ \min\{ x_{2j} : x_{1j} = c \}, \qquad \min\{ x_{2j} \mid x_{1j} = c \} \qquad\text{or}\qquad \min_{x_{1j}=c} x_{2j} $$ gets the job done.