Conditioning over a variable and a function

Conditioning over a variable and a function

I have a question if the following relation on probabilities hold for
independent random variables?
$$P_{X \mid Y, G(Y)}(x_1)=P_{X \mid \{Y\}}(x_2)$$ where $G$ is not
necessarily invertible.
Also can we say $$P_{X \mid Y, H(Z)}(x_3)=P_{X \mid Y, \{Z\}}(x_4)$$ where
$H$ is not necessarily invertible.
Finally can we say $$P_{X \mid Y, U}(x_5)=P_{X \mid Y,\{V\}}(x_6)$$ where
$U=g(V)$ is not necessarily invertible.
The values $x_1,x_2 $ can be equal or related through some function. same
goes to $x_3, x_4$ and $x_5,x_6$.If the mappings $H$, $G$ are invertible
does the answer change? Please explain the answer or provide a reference
which I can find online.
My guess is they hold with $G$, $H$, $g$ being invertible or not.