Equivalence of Definitions of Matroid Rank Axioms

Theorem
Let $S$ be a finite set.

Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.

Condition 1
$\rho$ satisfies formulation 1 of the rank axioms:

Condition 2
$\rho$ satisfies formulation 2 of the rank axioms:

Condition 3
$\rho$ is the rank function of a matroid $M = \struct{S, \mathscr I}$.

Condition 1 implies Condition 3
Let $\rho$ satisfy formulation 1 of the rank axioms:

Condition 3 implies Condition 2
Let $\rho$ be the rank function of a matroid $M = \struct{S, \mathscr I}$.

Condition 2 implies Condition 1
Let $\rho$ satisfy formulation 2 of the rank axioms: