Theorem petlemi

Description: If you can prove disjointness (e.g. disjALTV0 , disjALTVid , disjALTVidres , disjALTVxrnidres , search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV ), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. (Contributed by Peter Mazsa, 18-Sep-2021)

	Ref	Expression
Hypothesis	petlemi.1	$$⊢DisjR$$
Assertion	petlemi	$$⊢\left(DisjR\wedge \mathrm{dom}R/R=A\right)\leftrightarrow \left(EqvRel\wr R\wedge \mathrm{dom}\wr R/\wr R=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		petlemi.1	$$⊢DisjR$$
2	1	a1i	$$⊢\left(EqvRel\wr R\wedge \mathrm{dom}\wr R/\wr R=A\right)\to DisjR$$
3	2	petlem	$$⊢\left(DisjR\wedge \mathrm{dom}R/R=A\right)\leftrightarrow \left(EqvRel\wr R\wedge \mathrm{dom}\wr R/\wr R=A\right)$$