Why isn't an injection defined with an iff?

Question

Suppose that $f:X \rightarrow Y$ is a function.

Then an injection can be defined as:

$$\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Rightarrow x_1=x_2$$

Why isn't it defined instead as follows?

$$\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Leftrightarrow x_1=x_2$$

I think the above statement also captures the situation described by the definition of an injection in words, which is:

If no element of $Y$ is assigned to more than one elementof $X$, i.e. the function takes a different value for each point of the domain.

I can see that the definition is missing the phrase "if and only if". So is that the only reason we don't write "$\Leftrightarrow$"?

Answer 1

Recall that the fact that $f$ is a function implies that $x_1=x_2\implies f(x_1)=f(x_2)$.

So if you say that $f$ is an injective function, then $\leftarrow$ part is already true. So in order to save ourselves trivialities, we only state the $\implies$ part, and then when you want to prove that $f$ is injective you don't have to worry about that.

Answer 2

Both of the definition you propose are equivalent if $f$ is function. By definition if $f$ is a function, $x_1=x_2$ implies $f(x_1)=f(x_2)$.

Answer 3

If you suppose $f$ as a function, than the converse of your implication is not necessary. The implication $x_1=x_2 \implies f(x_1)=f(x_2)$ tells us only that f is a function!