How do I notate "the x-coördinate of $G^{-1}(z)$ for $z = G(x, y) : \mathbb{N}\times\mathbb{N} \to \mathbb{N}$

Question

My question is pretty much as given in the title, I have a function that bijectively maps $\mathbb{N}$ to $\mathbb{N}\times\mathbb{N}$, and I now wish to define a function which I, as a part-time programmer would write as $$K(x) = 2^{G^{-1}(z).x}\cdot 3^{G^{-1}(z).y}$$ However, this clearly cannot be mathematically correct since in mathematics maps do not return named tuples, they return ordered sets, so I could write $$(x, y) = G^{-1}(z) $$ but also $$(y, x) = G^{-1}(z) $$ or even $$(\mu, \aleph) = G^{-1}(z) $$ In what mathematical way could I write that I want two raised to a power equal to the variable that would have been first in ordering when taking $G(x, y) = z$ multiplied by three raised to a power equal to the parameter that would have been second in order when taking $G(x, y) = z$?

Answer 1

In general, when you have a function to a direct product $f : Z \to X \times Y$, it is common to denote $f_1 : Z \to X$ and $f_2 : Z \to Y$ the components, so that $f(z) = (f_1(z), f_2(z))$. So you could write $(G^{-1})_1(z)$ instead of $G^{-1}(z).x$.

Another possibility is to give a name to the projections $X \times Y \to X$ and $X \times Y \to Y$. A common choice is $\pi_1, \pi_2$ or $p_1, p_2$. And then $G^{-1}(z).x$ becomes $(\pi_1 \circ G^{-1})(z)$. Or omitting the symbol for function composition, $\pi_1 G^{-1}(z)$.