Proof of Upper Bounds of Elements of [0,∞], and the Graph of a Section of a Map on ℕ×X

Proof of (i). Let $c\in C$ with $c\in[0,\infty]$. By Order and Suprema in the Extended Real Line (viii), $0\le c$, and $c\le s$ by the hypothesis on $s$, so $0\le s$ by transitivity of the total order $\le$ (Order and Suprema in the Extended Real Line (i)). As $s\in[-\infty,\infty]$, Order and Suprema in the Extended Real Line (viii) gives $s\in[0,\infty]$.

Proof of (ii). Let $y\in X$ and let $v$ be a set with $(y,v)\in g$. Then $v=g(y)=w(n,y)$, and since $(n,y)\in\mathbb{N}\times X$, the pair $((n,y),w(n,y))=((n,y),v)$ lies in $w$ (by Maps between Classes: Values, Pairs, Extensionality, Restrictions, Constant Maps and Preimages (ii) for $g$, Maps between Classes: Values, Pairs, Extensionality, Restrictions, Constant Maps and Preimages (viii), and Maps between Classes: Values, Pairs, Extensionality, Restrictions, Constant Maps and Preimages (i) for $w$).