Denote there are two index sets $I= \{1, 2, \cdots, m\}$ and $J = \{1, 2, \cdots, n\}.$ Then, I have independent random variables $X_{ij}, \forall i \in I, j \in J.$ Fix $i \in I,$ we have $X_{ij}$ is identical for all $j \in J.$ 

Fix $k \in J,$ I am wondering if I could have the following probability inequality$$ P\left(X_{1k} \leq Y, X_{2k} \leq Y \mid Y \geq \max_{I\in I} \min_{j \in J} X_{ij}\right) \leq P\left(X_{1k} \leq Y \mid Y \geq \max_{I\in I} \min_{j \in J} X_{ij}\right) \cdot P\left( X_{2k} \leq Y \mid Y \geq \max_{I\in I} \min_{j \in J} X_{ij}\right)$$ 

If the inequality holds, how to show it? If the inequality does not hold, can we have some inequality of similar format? Thank you very much!