I'm reading the proof of the statement in the title in Picado and Pultr's book Frames and Locales (Proposition 7.2).
The authors first obtain the formula
$x\wedge \phi(a) = y\wedge \phi(a)\iff f^*(x)\wedge a = f^*(y)\wedge a.$
This is clear enough. However, they go on to say that this can obviously be rewritten as
$x\wedge \phi(a) \leq y\wedge \phi(a)\iff f^*(x)\wedge a \leq f^*(y)\wedge a.$
I'm not seeing why this is true. What am I missing? Thanks in advance for any helpful hints.
Here $f^*$ is the left adjoint to an open localic map $f:L\to M$, and $\phi$ is the map we get by noticing that an element $a\in L$ generates a sublocale $S$ of $L$, and the corresponding open sublocale $f[S]$ of $M$ is generated by an element of $M$, which we call $\phi(a)$ (using the fact that $f$ is assumed to be open).