Theorem eqvreldisj5

Description: Range Cartesian product with converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020) (Revised by Peter Mazsa, 22-Sep-2021)

	Ref	Expression
Assertion	eqvreldisj5	$$⊢EqvRelR\to DisjS⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_{\left(B/R\right)}\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqvreldisj3	$$⊢EqvRelR\to Disj\left({{\mathrm{E}}^{-1}\upharpoonright }_{\left(B/R\right)}\right)$$
2		disjimxrn	$$⊢Disj\left({{\mathrm{E}}^{-1}\upharpoonright }_{\left(B/R\right)}\right)\to DisjS⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_{\left(B/R\right)}\right)$$
3	1 2	syl	$$⊢EqvRelR\to DisjS⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_{\left(B/R\right)}\right)$$