Set Equivalence Less One Element

Theorem
Let $S$ and $T$ be sets such that $S \sim T$, i.e. they are equivalent.

Let $a \in S, b \in T$.

Then:
 * $S \setminus \left\{{a}\right\} \sim T \setminus \left\{{b}\right\}$

where $\setminus$ denotes set difference.

Proof
As $S \sim T$, there exists a bijection $f: S \to T$.

We define the mapping $g: \left({S \setminus \left\{{a}\right\}}\right) \to \left({T \setminus \left\{{b}\right\}}\right)$ as follows:

$\forall x \in S \setminus \left\{{a}\right\}: g \left({x}\right) = \begin{cases} f \left({x}\right): f \left({x}\right) \ne b \\ f \left({a}\right): f \left({x}\right) = b \end{cases}$

$g$ is shown to be a bijection as follows:

As $f$ is injective, we have that:
 * $\forall x_1, x_2 \in S: x_1 \ne x_2 \implies f \left({x_1}\right) \ne f \left({x_2}\right)$

Hence
 * $\forall x_1, x_2 \in S \setminus \left\{{a}\right\}: x_1 \ne x_2 \implies f \left({x_1}\right) \ne f \left({x_2}\right)$

It follows that:
 * $\forall x_1, x_2 \in S \setminus \left\{{a}\right\}: x_1 \ne x_2 \implies g \left({x_1}\right) \ne g \left({x_2}\right)$

and so $f$ is injective.


 * $\forall y \in T: \exists x \in S: f \left({x}\right) = y$

as $f$ is surjective.

So, all elements of $T \setminus \left\{{b}\right\}$ have images under $g$ which are the same as under $f$.

Except, that is, for the element whose preimage under $f$ is $a$.

Instead, this element is mapped to from the preimage of $b$ under $f$.

Hence we have that:
 * $\forall y \in T \setminus \left\{{b}\right\}: \exists x \in S \setminus \left\{{a}\right\}: y = f \left({x}\right)$

Thus $g \left({x}\right)$ is surjective.

So, being both injective and surjective, $g$ is a bijection.