Deformable Spaces In Spanier's Algebraic Topology, he states the following lemma:
"A space $X$ is deformable into a subspace A if and only if the inclusion map $i:A\hookrightarrow X$ has a right homotopy inverse."
Here is the proof he gives going from the second statement to the first:
"If $i$ has a right homotopy inverse $f:X\rightarrow A$, then $i\circ f \simeq 1_{X}$. Let $F:X\times I \rightarrow X$ be a homotopy from $1_{X}$ to  $i\circ f$; then $F(x,0) = x$, so $F$ is a deformation of $X$, and $F(X\times 1) = i\circ f(X) \subset A$, so $X$ is deformable into $A$."
My question is why is $i\circ f(X)$ a subset of A? Is it just by definition of the inclusion map or something else?

$f$ maps $X$ into $A$ and $i$ is just the identity on $A$. It is a trivial consequence of the definitions of $i$ and $f$.
