Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $A$, $B$ and $C$ be classes.

¶ If $F$ and $G$ are functions with equal domains $\operatorname{dom}F=\operatorname{dom}G$ and equal values $F(u)=G(u)$ for every set $u\in\operatorname{dom}F$, then $F=G$.

¶ If $F:A\to B$ and $G:B\to C$ are maps, then their composition is a map $G\circ F:A\to C$ and $(G\circ F)(u)=G(F(u))$ for every set $u\in A$.

¶ The identity $\mathrm{id}_{A}$ is a bijective map $\mathrm{id}_{A}:A\to A$ with $\mathrm{id}_{A}(u)=u$ for every set $u\in A$.

¶ Every map $F:A\to B$ satisfies $F\circ\mathrm{id}_{A}=F$ and $\mathrm{id}_{B}\circ F=F$.

¶ Every map $F:A\to B$ has its range equal to the image $\operatorname{ran}F=F[A]$; if moreover $F$ is injective, then its inverse is a bijective map $F^{-1}:\operatorname{ran}F\to A$ with $F^{-1}(F(u))=u$ for every set $u\in A$.

¶ If $F:A\to B$ is bijective, then its inverse is a bijective map $F^{-1}:B\to A$, and $F^{-1}\circ F=\mathrm{id}_{A}$ and $F\circ F^{-1}=\mathrm{id}_{B}$.

¶ If $F:A\to B$ and $G:B\to A$ are maps satisfying $G\circ F=\mathrm{id}_{A}$ and $F\circ G=\mathrm{id}_{B}$, then $F$ is bijective and $G=F^{-1}$.

¶ Let $F:A\to B$ and $G:B\to C$. If $F$ and $G$ are both injective (respectively both surjective, both bijective), then $G\circ F:A\to C$ is injective (respectively surjective, bijective).

¶ If $F$ is a function, then the restriction $F|_{A}$ is a function, its domain is the intersection $\operatorname{dom}(F|_{A})=A\cap\operatorname{dom}F$, and $(F|_{A})(u)=F(u)$ for every set $u\in A\cap\operatorname{dom}F$.