Theorem disjxrnres5

Description: Disjoint range Cartesian product. (Contributed by Peter Mazsa, 25-Aug-2023)

	Ref	Expression
Assertion	disjxrnres5	$$⊢DisjR⋉\left({S\upharpoonright }_A\right)\leftrightarrow \forall u\in A\forall v\in A\left(u=v\vee \left[u\right]R⋉S\cap \left[v\right]R⋉S=\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		xrnres2	$$⊢{R⋉S\upharpoonright }_A=R⋉\left({S\upharpoonright }_A\right)$$
2	1	disjeqi	$$⊢Disj\left({R⋉S\upharpoonright }_A\right)\leftrightarrow DisjR⋉\left({S\upharpoonright }_A\right)$$
3		xrnrel	$$⊢\mathrm{Rel}R⋉S$$
4		disjres	$$⊢\mathrm{Rel}R⋉S\to \left(Disj\left({R⋉S\upharpoonright }_A\right)\leftrightarrow \forall u\in A\forall v\in A\left(u=v\vee \left[u\right]R⋉S\cap \left[v\right]R⋉S=\varnothing \right)\right)$$
5	3 4	ax-mp	$$⊢Disj\left({R⋉S\upharpoonright }_A\right)\leftrightarrow \forall u\in A\forall v\in A\left(u=v\vee \left[u\right]R⋉S\cap \left[v\right]R⋉S=\varnothing \right)$$
6	2 5	bitr3i	$$⊢DisjR⋉\left({S\upharpoonright }_A\right)\leftrightarrow \forall u\in A\forall v\in A\left(u=v\vee \left[u\right]R⋉S\cap \left[v\right]R⋉S=\varnothing \right)$$