This might be a silly question, but I recently was confused by some notation regarding conditional probabilities. Suppose for any $y$ (for the sake of simplicity say this is a natural number) we are given an (non-deterministic) event $A[y]$. Now lets look at a random variable $Y$ (say $\sim\operatorname{Poi}(\lambda)$ for some $\lambda >0$). Is it true that \begin{equation} \mathbb{P}(A[Y] \mid Y=y) = \mathbb{P}(A[y])? \end{equation}
Heuristically this seems to be true, since the left hand side is the probability that "$A[Y]$ occurs under the condition, that "$Y=y$", which is also the quantity on the right, right?
I see people using this kind of notation when applying the total law of probability, like this \begin{equation} \mathbb{P}(A[Y]) = \sum_{y\in\mathbb{N}}\mathbb{P}(A[y])\cdot\mathbb{P}(Y=y), \end{equation} which confuses me, since, rigorously, there should be a conditional probability here.
Can someone provide a proof/counterexample?