It's known that ZF proves that for every set $x$ there exists a set $H_x$ of all sets that are hereditarily strictly subnumerous to $x$. [see here]. Now is the following principle equivalent with choice over the rest of axioms of ZF?

For every set $x$, there exists a set $y$ such that: $x$ is subnumerous to $H_y$.

By "subnumerous to" its meant, as usual, possessing an injection towards; and "strictly subnumerous" means, as usual, existence of subnumerousity without existence of a bijection.

It's known that ZF proves that for every set $x$ there exists a set $H_x$ of all sets that are hereditarily strictly subnumerous to $x$. [see here]. Now is the following principle equivalent with choice over the rest of axioms of ZF?

Every set $x$ is subnumerous to $H_x$.

By "subnumerous to" its meant, as usual, possessing an injection towards; and "strictly subnumerous" means, as usual, existence of subnumerousity without existence of a bijection.