Cardinality of Power Set is Invariant

Theorem
Let $X, Y$ be sets.

Let $\card X = \card Y$

where $\card X$ denotes the cardinality of $X$.

Then:
 * $\card {\powerset X} = \card {\powerset Y}$

where $\powerset X$ denotes the power set of $X$.

Proof
By definition of cardinality:
 * $X \sim Y$

where $\sim$ denotes the set equivalence.

Then by definition of set equivalence:
 * there exists a 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 direct image mapping $\map {f^\to}: \powerset X \to \powerset Y$ is an injection.

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

Then by definition of bijection:
 * $f^\to: \powerset X \to \powerset Y$ is a bijection.

Hence by definition of set equivalence:
 * $\powerset X \sim \powerset Y$

Thus the result by definition of cardinality:
 * $\card {\powerset X} = \card {\powerset Y}$