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Metamath Proof Explorer

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	⊢ Disj 𝑅
	Assertion	petlemi	⊢ ( ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ↔ ( EqvRel ≀ 𝑅 ∧ ( dom ≀ 𝑅 / ≀ 𝑅 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		petlemi.1	⊢ Disj 𝑅
2	1	a1i	⊢ ( ( EqvRel ≀ 𝑅 ∧ ( dom ≀ 𝑅 / ≀ 𝑅 ) = 𝐴 ) → Disj 𝑅 )
3	2	petlem	⊢ ( ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ↔ ( EqvRel ≀ 𝑅 ∧ ( dom ≀ 𝑅 / ≀ 𝑅 ) = 𝐴 ) )