Bijection iff Left and Right Inverse

Theorem
Let $$f: S \to T$$ be a mapping.

$$f$$ is a bijection iff:


 * $$\exists g_1: T \to S: g_1 \circ f = I_S$$
 * $$\exists g_2: T \to S: f \circ g_2 = I_T$$

where both $$g_1$$ and $$g_2$$ are mappings.

Proof

 * From Left Inverse Mapping, $$f$$ is an injection iff $$\exists g_1: T \to S: g_1 \circ f = I_S$$.


 * From Right Inverse Mapping $$f$$ is a surjection iff $$\exists g_2: T \to S: f \circ g_2 = I_T$$


 * If $$f$$ is a bijection, then it is both an injection and a surjection, thus both the described $$g_1$$ and $$g_2$$ must exist from Left Inverse Mapping and Right Inverse Mapping.

If both the described $$g_1$$ and $$g_2$$ exist, then it follows that $$f$$ is both an injection and a surjection, and therefore a bijection.