Little detail when finding the distribution of joint distribution Poisson r.v.'s, given the sum of these Poisson r.v.'s

Question

My professor did an exercise and I almost understood it entirely, but there's a little detail that I could not understood. This is the exercise:

We have 4 Poisson distributed random variables: $N_{11}, N_{12}, N_{21}, N_{22}$, with $N_{ij} \sim Poisson(\mu_{ij})$, all independent.
Make $N = \sum_{i, j = 1} ^ 2 N_{ij}$.
So we know that $N$ is also Poisson distributed, and $N \sim Poisson(\mu_{11} + \mu_{12} + \mu_{21} + \mu_{22})$.
Show that $M = N_{11}, N_{12}, N_{21}, N_{22} \ | \ N$ is Multinomial distributed.

I could understood all the calculations, and show that this is truly Multinomial distributed, but I could not understood the expression after the * flagged below:

$$P(N_{11} = n_{11}, N_{12} = n_{12}, N_{21} = n_{21}, N_{22} = n_{22} \ | \ N = n) = \\ \frac{P((N_{11} = n_{11}, N_{12} = n_{12}, N_{21} = n_{21}, N_{22} = n_{22}) \ \cap \ (N = n))}{P(N = n)} \stackrel{*}{=} \\ \frac{P(N_{11} = n_{11}, N_{12} = n_{12}, N_{21} = n_{21}, N_{22} = n - (n_{11} + n_{12} + n_{21}))}{P(N = n)} = \dots = \\ \frac{n!}{n_{11}! n_{12}! n_{21}! [n - (n_{11} + n_{12} + n_{21})]!} (p_{11} ^ {n_{11}}) (p_{12} ^ {n_{12}}) (p_{21} ^ {n_{21}}) (p_{22} ^ {n + (n_{11} - n_{12} - n_{21})}) $$
So we can see that $M \sim Multi(N, p_{11}, p_{12}, p_{21}, p_{22})$

The passage flagged with * is where I don't understood. What's the intuition about this passage?
I understood that $n_{22} = n - [n_{11} + n_{12} + n_{21}]$, but I'm confused about this passage.

This exercise is very similar with this one Conditional Distribution of Poisson Variables, given $\sum X_i$, but my doubt is this little detail that I showed in the passage flagged with *.

Answer 1

I think I know what's going on here.
This theorem resolves the problem: "Intersection with Subset is Subset"
In other words:
$$S \subseteq T \iff S \cap T = S $$

In our example, we have: $$ \{ N_{22} = n_{22} \} \subseteq \{ N = n \} \iff \{ N_{22} = n_{22} \} \cap \{ N = n \} = \{ N_{22} = n_{2} \}$$

That's why the intersection "dissappeared" and $\{ N_{22} = n_{22} \} = \{ N_{22} = n - (n_{11} + n_{12} + n_{21} + n_{22}) \}$ is maintained.

Here's an ugly draft: