Dirac Notation and Coordinate transformation of a function

Lets say we have a 1 dimensional system with coordinate $x$ and the associated operator $hat x$ with eigenstates $|xrangle$. A function of $x$ is defined as
$$
f(x) = langle x |frangle tag{1}
$$
Now we introduce a new coordinate $y$ with operator $hat y $ and corresponding eigenstates. x and y are describing the same 1 dimensional space. and both sets of eigenstates are assumed to be complete and orthogonal in the the sense of
$$
hat 1 = int dx ~|xrangle langle x|
langle x|x’rangle = delta(x-x’).tag{2}
$$

The relationship between coordinates $x$ and $y$ is given by
$$begin{aligned}x(y) &= ay
y(x) &= a^{-1}x
end{aligned}tag{3}$$
where $a$ is a real constant.

Due to completeness the function $f(x)$ can be expressed as
$$
f(x) = int ^infty_{-infty}dy langle x|yrangle langle y|frangletag{4}
$$

What is $langle x|yrangle$ in this case? What is the relationship of $|xrangle$ and $|yrangle$? How would all of this apply to an example function like $f(x)= x^2$?

Does the coordinate transformation define these relationships or do i need more definitions to answer these questions?

I tried the following
$$
f(x(y)) = tilde f(y) = a^2y^2 = langle y|frangle
x^2 = int ^infty_{-infty}dy langle x|yrangle a^2 y^2
Rightarrow langle x | y rangle = a^{-2}delta(x-y)tag{5}
$$
but that can’t be right. The result depends on the function that i plug in, which shouldn’t be the case. If had used a different $f(x)$ i would have gotten a different result for the matrix element $langle x |yrangle$. Where do i go wrong ?

My goal is to properly understand how a variable substitution, done in position representation, is expressed in Dirac Notation. Preferably starting in position representation and going "backwards" to Dirac Notation. I am uncertain how such a transformation is properly expressed in the more abstract Dirac representation or how it affects the states. I have no trouble doing a coordinate substitution in a integral of the form
$$
langle f|frangle = int f^*(x)f(x) dx tag{6}
$$
for the right side of the equation which is given in position representation but i fail at doing the same in a more abstract manner.

The right side simply transforms to
$$begin{aligned}
int langle f|xrangle langle x|frangle dx &= int f^*(x(y))f(x(y)) dx(y)
? &=int f^*(y) f(y) |frac{dx}{dy}| dy
end{aligned}tag{7}$$
On the other hand i can write the integral also as the following, using the completeness of $y$,
$$begin{aligned}
langle f | f rangle &= int langle f | y rangle langle y | f rangle dy
end{aligned}tag{8}$$
Now i set both equations expressed in $y$ equal,
$$
int f^*(y) f(y) |frac{dx}{dy}| dy = int langle f | y rangle langle y | f rangle dytag{9}
$$
which leads me to a contradiction, under the assumption that $f(y)=langle y | f rangle = f(x(y))$. Is the error the assumption
$$
f(y)=langle y | f rangle = f(x(y)) tag{10}
$$
or something else ?