Definition
A set $R$ is a (binary) relation if any its element is a ordered pair, that is $z\in R$ if and only if there exist $x$ and $y$ such that $z=(x,y)$. In particular if $R$ is contained in the cartesian product of two set $A$ and $B$ we say that $R$ is a relation of between $A$ and $B$ and if $B$ is equal to $A$ we say that $R$ is a relation in $A$.
Definition
If $R$ is a relation we call domain of $R$ that set whose elements are a first coordinate of some pair of the relation, that is $$ \text{dom}\,R:=\{x:\exists\,y\,\text{such that}\,(x,y)\in R\} $$ and analogously the range of $R$ is that set whose elements are the second coordinate of the relation, that is $$ \text{rank}\,R:=\{y:\exists\,x\,\text{such that}\,(x,y)\in R\} $$
So it is possible to prove that the above two defined set exist using the $ZFC$'s formalism but this now has not matter so we proceed to give the following well know definition.
Definition
A function $f$ is a relation such that if $(x,y)$ and $(x,z)$ are such that $(x,y),(x,z)\in f$ then $y=z$.
Definition
Let $A$ and $B$ sets. So the set whose element are functions from $A$ to $B$ is denoted by the symbol $B^A$.
Again it is possible to prove that the above defined set exist using $ZFC$'s formalism but now this again has not matter.
So with the previous definition I ask explain if it is true that the inclusion $$ A\subseteq B $$ implies the inclusion $$ B^C\subseteq A^C $$ for any other set $C$ as it seems Munkres stated. So could someone help me, please?