Theorem dfdisjs3

Description: Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	dfdisjs3	$$⊢\mathrm{Disjs}=\left\{r\in \mathrm{Rels}|\forall u\forall v\forall x\left(\left(urx\wedge vrx\right)\to u=v\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfdisjs2	$$⊢\mathrm{Disjs}=\left\{r\in \mathrm{Rels}|\wr r^{-1}\subseteq \mathrm{I}\right\}$$
2		cosscnvssid3	$$⊢\wr r^{-1}\subseteq \mathrm{I}\leftrightarrow \forall u\forall v\forall x\left(\left(urx\wedge vrx\right)\to u=v\right)$$
3	1 2	rabbieq	$$⊢\mathrm{Disjs}=\left\{r\in \mathrm{Rels}|\forall u\forall v\forall x\left(\left(urx\wedge vrx\right)\to u=v\right)\right\}$$