Notation for Coordinate when $x$ is a certain number, max $y$

Question

I need your help for expression for mathematical notation.

There is a set of $(x,y)$ points. Among points satisfying $x$ is a certain number in the set, I want to express min $y$ value.

For example, points set $= \{(1,2), (2,4), (1,5),(1,8)\}$ Points satisfying '$x = 1$' are $(1,2),(1,5)$ and $(1,8)$. So min $y$ is $2$.

How can I express them as mathematical notation? Please help me.

Thank you

Answer 1

Let $S$ be your set then you could use either of

min {$y|(x,y)\in S,x=1$}

min {$y|(1,y)\in S$}.

You can put whatever conditions you want to the right of the | symbol.