No Injection from Power Set to Set/Lemma

Theorem

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then there does not exist a set $B$ such that there is an injection from $B$ into $S$ and a surjection from $B$ onto $\powerset S$.

Proof

Aiming for a contradiction, suppose there exists such a $B$.

Let $i: B \to S$ be an injection.

Let $f: B \to \powerset S$ be a surjection.

Let $i^\gets: \powerset S \to \powerset B$ be the inverse image mapping of $i$.

By Mapping Induced by Inverse of Injection is Surjection, $i^\gets$ is a surjection.

Let $f^\to: \powerset B \to \powerset {\powerset S}$ be the direct image mapping of $f$.

By Direct Image Mapping of Surjection is Surjection, $f^\to$ is a surjection:

We have that $i^\gets: \powerset S \to \powerset B$ and $f^\to: \powerset B \to \powerset {\powerset S}$ are surjective.

By Composite of Surjections is Surjection, their composition $f^\to \circ i^\gets: \powerset S \to \powerset {\powerset S}$ is a surjection by Composite of Surjections is Surjection.

But this violates Cantor's Theorem, contradicting the assumption that such a $B$ exists.

$\blacksquare$