Proof of The Families of Rays of the Real Line Are Sets of Subsets of the Real Line

Take for the form $I(a,b)$ of Intervals of the Real Line as Defined Set Symbols, Typed Terms and Values of Maps the ray form of $R$: if $R(a)$ is $(a,\infty)$ or $[a,\infty)$, then $I(a,b)$ is $(a,\infty)$, respectively $[a,\infty)$; if $R(a)$ is $(-\infty,a)$ or $(-\infty,a]$, then $I(a,b)$ is $(-\infty,b)$, respectively $(-\infty,b]$. In each case $R(a)=I(a,a)$ for every $a\in\mathbb{R}$.

The identity $\mathrm{id}_{\mathbb{R}}$ is a map $\mathbb{R}\to\mathbb{R}$ with $\mathrm{id}_{\mathbb{R}}(a)=a$ for every $a\in\mathbb{R}$, by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (iii). By Intervals of the Real Line as Defined Set Symbols, Typed Terms and Values of Maps (iii), with $D=\mathbb{R}$ and $s=t=\mathrm{id}_{\mathbb{R}}$, there is a map $F:\mathbb{R}\to\mathcal{P}(\mathbb{R})$ with $F(a)=I(a,a)=R(a)$ for every $a\in\mathbb{R}$. Let $\mathcal{R}=F[\mathbb{R}]$ be the image of $\mathbb{R}$ under $F$; it is a set by Images of Sets under Functions Are Sets, and a Function with a Set Domain Is a Set (i). By The Image and the Preimage of a Class under a Class (i), a set $z$ lies in $\mathcal{R}$ if and only if $(a,z)\in F$ for some $a\in\mathbb{R}$, that is, $z=F(a)=R(a)$ for some $a\in\mathbb{R}$. So $\mathcal{R}$ has the stated property, and since every $F(a)$ lies in $\mathcal{P}(\mathbb{R})$, $\mathcal{R}\subseteq\mathcal{P}(\mathbb{R})$. Any set with the stated property has the same elements as $\mathcal{R}$ and so equals $\mathcal{R}$. Taking for $R$ each of the four forms in turn gives the last sentence of (i).