I just wanted a quick reality check from the SE community to make sure that I am correctly concluding the distinction between $G(f_{|0})$ and $G(f_{|1})$...where, in English, $f_{|a}$ is saying "the function $f$ restricted to domain $a$". For clarity, this question is taking place in the context of ordinals (i.e. $0 \in 1$ and $0$ is the least element of the ordinals).
I interpret $G(f_{|0})$ as being equivalent to $G(\emptyset)$ and $G(f_{|1})$ as being equivalent to $G(\{\langle\emptyset, x\rangle \})$, where the $x$ is whatever value $f(\emptyset)$ takes on.
Is this correct? Thank you! ~~