# security for PRF when key is changed often

Here's the situation. Suppose I have a secure PRF $F$ and use $F$ in the following way:

for equal size message $m_i$ in $\{m_j\}^{q}_{j=1}$( $q=poly(n)$, $n$ is security parameter ), randomly select $k_i \xleftarrow{\$}K$and compute$c_i=F(k_i, m_i)$, where$K$is key space. All previous computation results$\{(k_j, m_j, c_j)\}^{i-1}_{j=1}$are revealed to adversary$A$when using ($k_i$,$m_i$) to evaluate$F$. The main point is that the key is changed and some previous keys revealed, which is different to traditional pseudo-randomness definition of$PRF$. My security goal is adversary$A$cannot distinguish$c_i$from a random string$r_i$using any$PPT$distinguisher$D$. Intuitively, I think$F\$ is secure in this setting. Is there any formal security model for this situation? and under what situation is it secure or insecure?