ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4630 on D326: Cartesian product
Bijection between parenthesis-sliced cartesian triple products
Formulation 0
Let $X$, $Y$, and $Z$ each be a D11: Set such that
(i) $X \times Y \times Z$ and $(X \times Y) \times Z$ are each a D326: Cartesian product
(ii) \begin{equation} f : X \times Y \times Z \to (X \times Y) \times Z, \quad f(x, y, z) = ((x, y), z) \end{equation}
(iii) \begin{equation} g : (X \times Y) \times Z \to X \times Y \times Z, \quad g((x, y), z) = (x, y, z) \end{equation}
Then
(1) $f$ is a D468: Bijective map
(2) $g$ is an D216: Inverse map for $f$
Proofs
Proof 0
Let $X$, $Y$, and $Z$ each be a D11: Set such that
(i) $X \times Y \times Z$ and $(X \times Y) \times Z$ are each a D326: Cartesian product
(ii) \begin{equation} f : X \times Y \times Z \to (X \times Y) \times Z, \quad f(x, y, z) = ((x, y), z) \end{equation}
(iii) \begin{equation} g : (X \times Y) \times Z \to X \times Y \times Z, \quad g((x, y), z) = (x, y, z) \end{equation}
This result is a particular case of R1848: Bijection between parenthesis-sliced cartesian products. $\square$