I am looking to use randomized public-key encryption in a context where it should also serve as a sort of "binding commitment". That is, I want to encrypt a value $x$ with some randomness $rnd$ under a public key $pk$, denoted $enc(x, rnd, pk)$ for short, such that it is infeasible to find a different plaintext $x'$ and randomness $rnd'$ such that $enc(x, rnd, pk) = enc(x', rnd', pk)$ (the ciphertexts are the same).
I want this because a party A should be able to convince a party B that a given ciphertext $c$ is an encryption of a plaintext (known to the parties) $x$. For this, A would send the ciphertext $c$ and the randomness $rnd$ to B, and B would re-compute $enc(x, rnd, pk)$. When this equals $c$, then B should be convinced that $c$ is an encryption of $x$.
Is this a property that randomized public-key encryption schemes usually have, or can be derived from another property?