Equality of Elements in Range of Mapping

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

Then:


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

Proof

 * Let $$f \left({x_1}\right) = f \left({x_2}\right)$$.

Then:

$$ $$


 * Now let $$\exists y \in \mathrm{Rng} \left({f}\right): \left({x_1, y}\right) \in f \land \left({x_2, y}\right) \in f$$.

Then:

$$ $$ $$

The result follows from the definition of Material Equivalence.