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 is Injection iff Direct Image Mapping 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:

$\card {\powerset X} = \card {\powerset Y}$

$\blacksquare$

Sources

• Mizar article ZFREFLE1:12