Identifying a set $S$ with a subset in $T$. Let $S = \{1,2\} \subset \mathbb{N}$, and $T = \{5,6,7\} \subset \mathbb{N}$. Clearly, $S \not\subset T$.
Let $i:S \to \{5,6\}$ be such that $i(n) = n+4$. This is injective and surjective. So, $i$ is an isomorphism.
Then, can I "identify" $S$ with a subset of $T$ via $i$, and hence $S \subset T$? If yes, what is the mathematical machinary behind this? Are we using the equivalence relation "$n \sim i(n)$ for $n \in S$"? 
 A: There is no actual mathematics behind this type of convention.
Any suppressed identification (such as $\Bbb R \subseteq \Bbb C$ as in the comments) is an abuse of notation, which aids our intuition because we can guess (or "intuitively know", if you prefer) what the identification/inclusion map $i: \Bbb R \hookrightarrow \Bbb C$ is.
This kind of "abuse" makes our life easier because it allows us to obfuscate trivialities and focus on the interesting stuff at hand.
For example, if $F' \subseteq F$ is a subfield, then we will usually use $+$ for both addition in $F'$ and in $F$. But then if we write $f_1 + f_2$ for $f_1,f_2 \in F'$, do we mean $+: F' \times F' \to F'$ or $+: F \times F \to F$? Obviously, it "doesn't matter", so we drop the distinction.

If you are interested in this kind of thing, consider having a look at category theory. Through making this type of identification explicit, it uncovers a lot of interesting results applicable in all of mathematics. Have a look at the concept of a "forgetful functor".
