Short version: Suppose I have a function $$R^*=q(Z_i,\lambda p_mR_{-i},p_mp_t,p_m)$$, is it equivalent to change #/form of arguments and write this as $$f(Z_i,\lambda, p_m,R_{-i}, p_t)$$, or, if I suppose/require that $Z_i$ and $R_{-i}$ are dependent on some other variables $u,v$, can I write $$g(u,v,\lambda,p_m,p_t) \equiv q(Z_i,\lambda p_mR_{-i},p_mp_t,p_m)$$
Longer version (with some background) Consider the maximization problem $$ \max_{R} U(Z_i-p_mp_tR,p_m(R-\lambda R_{-i}),R)$$ the FOC is then $$\tag{1} -p_mp_tU_1(Z_i-p_mp_tR,p_m(R-\lambda R_{-i}),R) + p_mU_2(\cdot,\cdot,\cdot) +U_3(\cdot,\cdot,\cdot)=0$$
The source I come across this in says that the derivatives of $U$ in (1) depend on 4 arguments, $$Z_i\quad, p_m\lambda R_{-i},\quad p_mp_t,\quad p_m$$ and that, since each of these are "exogenous" in the problem (taken as given), we can write the solution to the maximization as $$R^*=q(Z_i,\lambda p_mR_{-i},p_mp_t,p_m)$$
My question is: Would it matter if I instead said that the derivatives in the FOC depend on 5 arguments, $Z_i\quad, R_{-i},\quad \lambda, \quad p_t,\quad p_m$ and wrote the solution as $$ R^* = f(Z_i,\lambda, R_{-i}, p_m,p_t)$$
Some explanation as to why it would/wouldnt matter would be appreciated.
(if necessary, just assume a solution exists to the maximization)