The Sets of Injections, Surjections and Bijections between Two Sets

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $A$ and $B$ be sets, and let $B^{A}$ be the set of maps from $A$ to $B$ (The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii), The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (i)).

(i) Injections. The set of injections from $A$ to $B$ is

$$\operatorname{Inj}(A,B)=\{f\in B^{A}:f\text{ is injective}\},$$

with injective as in Injective, Surjective and Bijective Functions between Classes (i). For a set $f$, $f\in\operatorname{Inj}(A,B)$ if and only if $f:A\to B$ is injective.

(ii) Surjections. The set of surjections from $A$ to $B$ is

$$\operatorname{Surj}(A,B)=\{f\in B^{A}:f\text{ is surjective}\},$$

with surjective as in Injective, Surjective and Bijective Functions between Classes (ii). For a set $f$, $f\in\operatorname{Surj}(A,B)$ if and only if $f:A\to B$ is surjective.

(iii) Bijections. The set of bijections from $A$ to $B$ is

$$\operatorname{Bij}(A,B)=\{f\in B^{A}:f\text{ is bijective}\},$$

with bijective as in Injective, Surjective and Bijective Functions between Classes (iii). For a set $f$, $f\in\operatorname{Bij}(A,B)$ if and only if $f:A\to B$ is bijective.