Do Randoms Cancel Out?

Let $A$ be a stochastic variables taking values on the set $\{1,\dots,10\}$ .

The definition of $A$ as "pick random between (pick random between 1 and 10) and (pick random between 1 and 10)" is awkward, but as we see it doesn't matter!

Let $X \sim u(1,\dots,10)$ be the uniform distribution on the same set, independent of $A$ . Now the probability asked for is $\mathbb P(X = A) = \sum_{n=1}^{10} \mathbb P(X = n \wedge A = n) \\ = \sum_{n=1}^{10} \mathbb P(X = n) \mathbb P(A = n) \\ = \sum_{n=1}^{10} \dfrac 1{10} \mathbb P(A = n) \\ = \dfrac 1{10} \sum_{n=1}^{10} \mathbb P(A = n) \\ = \dfrac 1{10} \cdot 1 = \boxed{\dfrac 1{10}}.$ Here we used respectively the independence of $A$ and $X$ , and the uniformity of $X$ .

While this problem may appear quite fiendish, it becomes quite simple with a key insight - it does not matter what the integer before the equals sign is!

While the probability of the integer some numbers is different to the probability of the integer being other numbers, this does not matter. As the first integer must be between $1$ and $10$ , the probability that the second integer is equal to it is $\frac{1}{10}$ .

I'm not completely sure this is rigourous. Could someone could confirm/deny that it is?