Is Y Contractible to a Point Proof for Homotopy Theory?

[CornMuffin]

Homework Statement

Prove that if Y is contractible, then any two maps from X to Y are homotopic.


I feel like I have a very, very, very sloppy proof :(

Homework Equations

The Attempt at a Solution

Assume Y is contractible to a point y₀ held fixed, then there is a map F:Y x [0,1] -> Y such that
(i) F(y,0)=y₀, for all y in Y
(ii) F(y,1)=y, for all y in Y
(iii) F(y₀,t)=y₀, for all t in [0,1]

Let F(y,t) be the homotopy of Y to a point y₀
Claim: any map f:X -> Y is homotopic to y₀ by the homotopy f_t(x)=F(f(x),t)
Proof of Claim: Since f(x)=y in Y, f_t=F(y,t), where y is in Y and t is in [0,1],
And since Y is contractible, f:X -> Y is homotopic to y₀

Therefore, any two maps from X to Y are homotopic.

 

[lanedance]

I don't know much about homotopies but you reasoning seems ok, it seems pretty clear that any map f is homotopic to the constant to y₀.

The only assumption I can see in there, maybe worth checking you can use, is that a homotopy of functions is an equivalence class, ie.
f ~ y₀
g ~ y₀
implies
f ~ g