Theorem dfdisjALTV5

Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV5 . (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	dfdisjALTV5	$$⊢DisjR\leftrightarrow \left(\forall u\in \mathrm{dom}R\forall v\in \mathrm{dom}R\left(u=v\vee \left[u\right]R\cap \left[v\right]R=\varnothing \right)\wedge \mathrm{Rel}R\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfdisjALTV2	$$⊢DisjR\leftrightarrow \left(\wr R^{-1}\subseteq \mathrm{I}\wedge \mathrm{Rel}R\right)$$
2		cosscnvssid5	$$⊢\left(\wr R^{-1}\subseteq \mathrm{I}\wedge \mathrm{Rel}R\right)\leftrightarrow \left(\forall u\in \mathrm{dom}R\forall v\in \mathrm{dom}R\left(u=v\vee \left[u\right]R\cap \left[v\right]R=\varnothing \right)\wedge \mathrm{Rel}R\right)$$
3	1 2	bitri	$$⊢DisjR\leftrightarrow \left(\forall u\in \mathrm{dom}R\forall v\in \mathrm{dom}R\left(u=v\vee \left[u\right]R\cap \left[v\right]R=\varnothing \right)\wedge \mathrm{Rel}R\right)$$