Zero information is exactly independence

By [F2] and [F1], $I(X;Y)=0$ exactly when $P_{XY}=P_X\otimes P_Y$, which is the stated independence condition.

By [F3] and [F1], $I(X;Y\mid Z)=0$ exactly when the joint law $P_{ZXY}$ of $(Z,X,Y)$ equals the conditional-product reference law R.

Suppose $P_{ZXY}=R$, using the coordinate order $(Z,X,Y)$ of [F3]. For all measurable A,B,C the rectangle formula for R gives $P(X\in A,Y\in B,Z\in C)={\int }_{C}K(z,A)L(z,B)P_Z(dz)\mathrm{.}$ By the regular conditional law identities, $K(Z,A)$ and $L(Z,B)$ are versions of the two conditional expectations in [F4]. The displayed identity says their product is the conditional-expectation version of ${\mathbf{1}}_{\{X\in A,Y\in B\}}$ by [F5]. Thus [F4] holds for indicators. Linearity gives it for bounded simple functions; bounded measurable functions follow by bounded simple approximation and [F6]. Conversely, [F4] applied to indicators gives the same rectangle identity, so $P_{ZXY}$ and R agree on every rectangle $C\times A\times B$. These rectangles determine the probability law by [F7], hence $P_{ZXY}=R$.