Free Module is Isomorphic to Free Module on Set

Theorem
Let $M$ be a unitary $R$-module.

Let $S=(m_i)_{i\in I}$ be a family of elements of $M$.

Let $\Psi:R^{(I)}\to M$ be the morphism given by Universal Property of Free Module Indexed by Set.

Then $S$ is a basis of $M$ iff $\Psi$ is an isomorphism.

Proof
Combine Characterisation of Linearly Independent Set through Free Module Indexed by Set and Characterisation of Spanning Set through Free Module Indexed by Set.