Show that in any map $f:X \rightarrow Y$, the map $F:X \rightarrow X \times Y$, defined by $x\mapsto^{F} (x, f(x))$ is injective.

Question

My attempt:

Let $x_1, x_2 \in X$, such that $x_1 \ne x_2$. Then $F(x_1)=(x_1, f(x_1))$ and $F(x_2)=(x_2, f(x_2))$. Since $x_1 \ne x_2$, $F(x_1) \ne F(x_2)$, irrespective of whether $f(x_1) = f(x_2)$ or not, so F is always injective.