On describing fcc and bcc planes in a Cartesian system (simple cubic system)

Ashcroft and Mermin remark, at the bottom of page 91 that

As a general rule, face-centered and body-centered cubic Bravais lattice[s] are described in terms of a conventional cubic cell, i.e. as simple cubic lattices with bases.

So far, no problem. They go on to say:

Since any lattice plane in a fcc or bcc lattice is also a lattice plane in the underlying simple cubic lattice, the same elementary cubic indexing can be used to specify lattice planes.

It is the bolded part which I am asking about -- I can convince myself intuitively that this should be true, but cannot prove it. Also, does the converse hold? Is every lattice plane in the underlying simple cubic lattice also a lattice plane in the given fcc or bcc structure? Essentially, I am asking for a proof that, given a fixed bcc or fcc lattice, there is a one-to-one correspondence between lattice planes as described in bcc or fcc and as described in the underlying simple cubic lattice.

Finally, they conclude with:

In practice, it is only in the description of noncubic crystals that one must remember that the Miller indices are the coordinates of the normal [to the lattice plane] in a system given by the reciprocal lattice, rather than the direct lattice [because the reciprocal of simple cubic is simple cubic].

The conclusion, I suppose, is thus that whenever we see Miller indices used to describe planes in fcc or bcc, by convention these are not in fact coordinates in the corresponding reciprocal lattices bcc and fcc, respectively (as they would be by the strict definition of Miller indices), but are instead coordinate in the reciprocal simple cubic lattice (to the underlying simple cubic direct lattice used to describe the crystal)? I can accept this only if the one-to-one correspondence asked about above exists.

• Contributors of Matter Modelling SE could provide a detailed answer if the question is displaced. The statement is intuitive and could be surely demontrated. You can start by checking if there are intergers $n_i$ so that two translational vectors $R=n_1 a_1 + n_2 a_2 + n_3 a_2$ hold for SC, BCC and FCC ($a_i$ are primitive vectors). May 17 at 23:36