An equidimensional $C^1$ map sends null sets to null sets

If $m=0$, then nullity means emptiness by [L1], so $E=\mathrm{\varnothing }$ and the claim follows. Assume $m\ge 1$. By [L0] under the stated Countable Choice, take countable chart atlases $(U_j,{\phi }_j)$ of $M$ and $(V_k,{\psi }_k)$ of $N$. For each pair $(j,k)$, the set $D_{jk}:={\phi }_j(U_j\cap F^{-1}(V_k))$ is open in ${\mathbb{R}}^m$ and the coordinate map $f_{jk}:={\psi }_k\circ F\circ {\phi }_j^{-1}$ is $C^1$ on $D_{jk}$.

For every $(j,k)$, enumerate the nondegenerate closed cubes with rational endpoints whose closures lie in $D_{jk}$. Their interiors cover $D_{jk}$, and the family is countable. For each such closed cube $Q$, choose an open convex cube $Q^{\mathrm{\prime }}$ with $Q\subset Q^{\mathrm{\prime }}$ and $\overline{Q^{\mathrm{\prime }}}\subset D_{jk}$. The derivative of $f_{jk}$ is bounded on $\overline{Q^{\mathrm{\prime }}}$ by [L2]. Apply the mean-value inequality on the open convex set $Q^{\mathrm{\prime }}$, and extend its estimate to $\overline{Q^{\mathrm{\prime }}}$ by continuity; in particular $f_{jk}{\mathrm{\mid }}_Q$ is Lipschitz. Define ${\pi }_Q:{\mathbb{R}}^m\to Q$ by clamping each coordinate to its interval; each scalar clamp is $1$-Lipschitz, so ${\pi }_Q$ is $1$-Lipschitz in Euclidean norm and fixes $Q$. Hence $f_{jk}\circ {\pi }_Q$ is Lipschitz on all of ${\mathbb{R}}^m$. The set $A_{jkQ}:={\phi }_j(E\cap U_j)\cap Q$ is null by [L1], so [L3] makes $f_{jk}(A_{jkQ})=(f_{jk}\circ {\pi }_Q)(A_{jkQ})$ null.

Fix $k$. Every point of ${\psi }_k(F(E)\cap V_k)$ comes from some $p\in E\cap U_j\cap F^{-1}(V_k)$, and ${\phi }_j(p)$ lies in the interior of one of the cubes from step 2.1. Thus ${\psi }_k(F(E)\cap V_k)$ is contained in the countable union of the null sets $f_{jk}(A_{jkQ})$ over $j,Q$. Countable subadditivity makes it null. Since this holds for each $k$, [L1] detects $F(E)$ as null in $N$.