I am studying the lectures on Motivic cohomology and I have a problem in the proof of Lemma 1.7.
The lemma says the following: Let $V\subset X\times Y$ and $W\subset Y\times Z$ be irreducible closed subsets which are finite and surjective over $X$ and $Y$ respectively ($X,Y,Z$ are smooth over a field $k$). Then $V\times Z$ and $X\times W$ intersect properly.
The definition of proper intersection given above the lemma is the following:
Definition: Two irreducible closed subsets $Z_{1},Z_{2}$ of a smooth scheme are said to intersect properly if $Z_{1}\cap Z_{2}=\emptyset$ or $\operatorname{codim}(Z_{1}\cap Z_{2})=\operatorname{codim}(Z_{1})+\operatorname{codim}(Z_{2})$
The lemma starts as follows. We consider the associated integral subschemes of $V$ and $W$ which are noted as $\tilde{V}$ and $\tilde{W}$ respectively. Then we assume without loss of generality that $X,Y$ are connected schemes (since in any case we would give a proof for connected components), so this means $X,Y$ are smooth and connected over a field, hence irreducible.
Next, we consider the fiber product $\tilde{V}\times_{Y}\tilde{W}$ and we prove by another lemma that it is finite and surjective over $\tilde{V}$ and therefore over $X$ too (I can understand why this argument is true). Then we argue that the induced map
$$\tilde{V}\times_{Y}\tilde{W}\to X\times Y\times Z$$ has image $T=\tilde{V}\times Z\cap X\times\tilde{W}$. Then we have that each irreducible component $T_{i}$ of $T$ is the image of an irreducible component of $\tilde{V}\times_{Y}\tilde{W}$ (I should understand it clearly but I think it is not my problem).
The last argument is that $T_{i}$ is finite and surjective over $X$, finite implies integral and by this Lemma the dimensions $\dim T_{i}=\dim X$ for all $i$ are the same. He then concludes that the closed subschemes $\tilde{V}\times Z$ and $X\times\tilde{W}$ intersect properly.
Namely, I cannot understand the (last) argument given for proving that the closed subschemes $\tilde{V}\times Z$ and $X\times\tilde{W}$ intersect properly. It might be a problem that the definition of proper intersection is given by codimensions (it is given in the lecture notes above the lemma). Any help is appreciated!
Key fact: if $f\colon A\to B$ is finite and surjective, then $\dim A=\dim B$.
We know that $\tilde{W}$ is finite and surjective over $X$. So $\dim \tilde{W}=\dim X$. This implies $\operatorname{codim} \tilde{W}\times Z = \dim X+\dim Y+\dim Z -(\dim \tilde{W}+\dim Z)=\dim X+\dim Y-\dim X = \dim Y.$
Similarly $\operatorname{codim} X\times \tilde{W} = \dim Z$. So we need to show that $\operatorname{codim} (X\times \tilde{W})\cap (\tilde{W}\times Z)=\dim Y+\dim Z.$
But each $T_i$ is finite and surjective over $X$ and so $\dim X=\dim T_i$. This implies $\operatorname{codim} T_i=\dim Y+\dim Z.$