Charge-Conjugation property

[BobaJ]

Homework Statement

I have to show, the last equality in the charge-conjugation property of the current $$C\bar{\Psi}_a\gamma^{\mu}\Psi_bC^{-1}=\bar{\Psi}_a^c\gamma^{\mu}\Psi_b^c=-(\bar{\Psi}_a\gamma^{\mu}\Psi_b)^\dagger$$

Relevant Equations

$\bar{\Psi}^c=-\Psi^TC^{-1}$

$\Psi^c=C\bar{\Psi}^T$

$C^{-1}\gamma^{\mu}C=-\gamma^{\mu T}$

$C=-C^{-1}=-C^\dagger=-C^T$

And I have to use that the fermionic quantum fields $\Psi_a$ and $\Psi_b$ anticommute.

I'm probably just complicating things, but I'm a little bit stuck with this problem.

I started with just plugging in the definitions for $\bar{\Psi}_a^c$ and $\Psi_b^c$. So I get

$$\bar{\Psi}_a^c\gamma^{\mu}\Psi_b^c=-\Psi_a^TC^{-1}\gamma^{\mu}C\bar{\Psi}_b^T$$.

After this I used $C^{-1}\gamma^{\mu}C=-\gamma^{\mu T}$ to get:

$$=\Psi-a^T\gamma^{\mu T}\bar{\Psi}_b^T$$.

Is this the right way? How do I go onto show the equality? Thank you for your help.