Equality of Elements in Range of Mapping

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

Then:


 * $\exists y \in \Rng f: \tuple {x_1, y} \in f \land \tuple {x_2, y} \in f \iff \map f {x_1} = \map f {x_2}$

Necessary Condition
Let:
 * $\exists y \in \Rng f: \tuple {x_1, y} \in f \land \tuple {x_2, y} \in f$

Then:

Sufficient Condition
Let:
 * $\map f {x_1} = \map f {x_2}$

Then:

The result follows from the definition of logical equivalence.