Cardinality of Power Set is Invariant

Theorem
Let $X, Y$ be [Definition:Set|sets]].

Let $\left\vert{X}\right\vert = \left\vert{Y}\right\vert$

where $\left\vert{X}\right\vert$ denotes the cardinality of $X$.

Then $\left\vert{\mathcal P \left({X}\right)}\right\vert = \left\vert{\mathcal P \left({Y}\right)}\right\vert$

where $\mathcal P \left({X}\right)$ denotes the power set of $X$.

Proof
By definition of [[Definition:Cardinality|cardinality]:
 * $X \sim Y$

where $\sim$ denotes the set equivalence.

Then by definition of set equivalence:
 * there exists a [[Definition:Bijection|bijection] $f: X \to Y$

By definition of bijection
 * $f$ is an injection and a surjection.

By Mapping Induced on Power Set by Injection is Injection:
 * the mapping induced $f^\to:\mathcal P \left({X}\right) \to \mathcal P \left({Y}\right)$ is an injection.

By Mapping Induced on Power Set by Surjection is Surjection
 * $f^\to$ is a surjection.

Then by definition of bijection:
 * $f^\to:\mathcal P \left({X}\right) \to \mathcal P \left({Y}\right)$ is a bijection.

Hence by definition of set equivalence:
 * $\mathcal P \left({X}\right) \sim \mathcal P \left({Y}\right)$

Thus the result by definition of cardinality:
 * $\left\vert{\mathcal P \left({X}\right)}\right\vert = \left\vert{\mathcal P \left({Y}\right)}\right\vert$