Equality of Elements in Range of Mapping

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

Then:


 * $\exists y \in \operatorname{Rng} \left({f}\right): \left({x_1, y}\right) \in f \land \left({x_2, y}\right) \in f \iff f \left({x_1}\right) = f \left({x_2}\right)$

Necessary Condition
Let:
 * $\exists y \in \operatorname{Rng} \left({f}\right): \left({x_1, y}\right) \in f \land \left({x_2, y}\right) \in f$

Then:

Sufficient Condition
Let:
 * $f \left({x_1}\right) = f \left({x_2}\right)$

Then:

The result follows from the definition of logical equivalence.