In regards to your comments, if you have two infinite theories $T$ and $T'$, if $T'$ consists of $T$ plus the schema $M\vDash T$ for some constant $M$, without alternate assumptions (Such as the existence of a truth predicate) $T'\nvdash Con(ZFC)$. This is why $ZFC+V_\kappa\prec V\nvdash V_\kappa\vDash ZFC$.

As for your actual theory, if $M\vDash ZFC$, then there is some $(N,V_\kappa^N)$ such that $V_\kappa^N\prec N$. I claim for any formula $\phi(x_0...x_n)\leftrightarrow\phi^{N_0}(x_0...x_n)$, where $N_0=\text{def}(N)$, if $\{(x_0...x_n)|\phi(x_0...x_n)\}\subseteq V_\kappa^N$, then $\{(x_0...x_n)|\phi(x_0...x_n)\}\in V_\kappa^N$ and $\{(x_0...x_n)|\phi(x_0...x_n)\}$ is definable in $V_\kappa^N$, the only tricky case being the existential quantifier.

Note that $\{(x_0...x_n)|\exists x(\phi(x,x_0...x_n))\}=\text{ran}(\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})$. Now if $\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})$ is definable in $V_\kappa^N$, then $\text{ran}(\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})\in V_\kappa^N$. Furthermore, $\{(x_0...x_n)|\exists x(\phi(x,x_0...x_n))\}$ is definable as $\text{ran}\{(x_0...x_n)|\phi(x,x_0...x_n)\}$. The rest is trivial.

In regards to your comments, if you have two infinite theories $T$ and $T'$, if $T'$ consists of $T$ plus the schema $M\vDash T$ for some constant $M$, without alternate assumptions (Such as the existence of a truth predicate) $T'\nvdash Con(T)$. This is why $ZFC+V_\kappa\prec V\nvdash V_\kappa\vDash ZFC$.

As for your actual theory, if $M\vDash ZFC$, then there is some $(N,V_\kappa^N)$ such that $V_\kappa^N\prec N$. I claim for any formula $\phi(x_0...x_n)\leftrightarrow\phi^{N_0}(x_0...x_n)$, where $N_0=\text{def}(N)$, if $\{(x_0...x_n)|\phi(x_0...x_n)\}\subseteq V_\kappa^N$, then $\{(x_0...x_n)|\phi(x_0...x_n)\}\in V_\kappa^N$ and $\{(x_0...x_n)|\phi(x_0...x_n)\}$ is definable in $V_\kappa^N$, the only tricky case being the existential quantifier.

Note that $\{(x_0...x_n)|\exists x(\phi(x,x_0...x_n))\}=\text{ran}(\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})$. Now if $\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})$ is definable in $V_\kappa^N$, then $\text{ran}(\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})\in V_\kappa^N$. Furthermore, $\{(x_0...x_n)|\exists x(\phi(x,x_0...x_n))\}$ is definable as $\text{ran}\{(x_0...x_n)|\phi(x,x_0...x_n)\}$. The rest is trivial.

In regards to your comments, Zuhair, if you have two infinite theories $T$ and $T'$, if $T'$ consists of $T$ plus the schema $M\vDash T$ for some constant $M$, without alternate assumptions (Such as the existence of a truth predicate) $T'\nvdash Con(ZFC)$. This is why $ZFC+V_\kappa\prec V\nvdash V_\kappa\vDash ZFC$.

As for your actual theory, if $M\vDash ZFC$, then there is some $(N,V_\kappa^N)$ such that $V_\kappa^N\prec N$. I claim for any formula $\phi(x_0...x_n)\leftrightarrow\phi^{N_0}(x_0...x_n)$, where $N_0=\text{def}(N)$, if $\{(x_0...x_n)|\phi(x_0...x_n)\}\subseteq V_\kappa^N$, then $\{(x_0...x_n)|\phi(x_0...x_n)\}\in V_\kappa^N$ and $\{(x_0...x_n)|\phi(x_0...x_n)\}$ is definable in $V_\kappa^N$, the only tricky case being the existential quantifier.

Note that $\{(x_0...x_n)|\exists x(\phi(x,x_0...x_n))\}=\text{ran}(\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})$. Now if $\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})$ is definable in $V_\kappa^N$, then $\text{ran}(\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})\in V_\kappa^N$. Furthermore, $\{(x_0...x_n)|\exists x(\phi(x,x_0...x_n))\}$ is definable as $\text{ran}\{(x_0...x_n)|\phi(x,x_0...x_n)\}$. The rest is trivial.

In regards to your comments, if you have two infinite theories $T$ and $T'$, if $T'$ consists of $T$ plus the schema $M\vDash T$ for some constant $M$, without alternate assumptions (Such as the existence of a truth predicate) $T'\nvdash Con(ZFC)$. This is why $ZFC+V_\kappa\prec V\nvdash V_\kappa\vDash ZFC$.

As for your actual theory, if $M\vDash ZFC$, then there is some $(N,V_\kappa^N)$ such that $V_\kappa^N\prec N$. I claim for any formula $\phi(x_0...x_n)\leftrightarrow\phi^{N_0}(x_0...x_n)$, where $N_0=\text{def}(N)$, if $\{(x_0...x_n)|\phi(x_0...x_n)\}\subseteq V_\kappa^N$, then $\{(x_0...x_n)|\phi(x_0...x_n)\}\in V_\kappa^N$ and $\{(x_0...x_n)|\phi(x_0...x_n)\}$ is definable in $V_\kappa^N$, the only tricky case being the existential quantifier.

Note that $\{(x_0...x_n)|\exists x(\phi(x,x_0...x_n))\}=\text{ran}(\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})$. Now if $\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})$ is definable in $V_\kappa^N$, then $\text{ran}(\{(x,x_0...x_n)|\phi(x,x_0...x_n)\})\in V_\kappa^N$. Furthermore, $\{(x_0...x_n)|\exists x(\phi(x,x_0...x_n))\}$ is definable as $\text{ran}\{(x_0...x_n)|\phi(x,x_0...x_n)\}$. The rest is trivial.