Injection iff Left Cancellable/Necessary Condition

Theorem

Let $f: Y \to Z$ be an injection.

From the definition: a mapping $f: Y \to Z$ is left cancellable if and only if:

$\forall X: \forall g_1: X \to Y, g_2: X \to Y: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$

Let $f: Y \to Z$ be an injection.

Let $X$ be a set

Let $g_1: X \to Y, g_2: X \to Y$ be mappings such that:

$f \circ g_1 = f \circ g_2$

Then $\forall x \in X$:

$\ds \map f {g_1 \paren x}$

$=$

$\ds \map {f \circ g_1} x$

$\ds $

$=$

$\ds \map {f \circ g_2} x$

$\ds $

$=$

$\ds \map f {\map {g_2} x}$

As $f$ is an injection, $\map {g_1} x = \map {g_2} x$ and thus the condition for left cancellability holds.

$\blacksquare$