Theorem dm1cosscnvepres

Description: The domain of cosets of the restricted converse epsilon relation is the union of the restriction. (Contributed by Peter Mazsa, 18-May-2019) (Revised by Peter Mazsa, 26-Sep-2021)

	Ref	Expression
Assertion	dm1cosscnvepres	⊢ dom ≀ ( ◡ E ↾ 𝐴 ) = ∪ 𝐴

Proof

Step	Hyp	Ref	Expression
1		dmcoss2	⊢ dom ≀ ( ◡ E ↾ 𝐴 ) = ran ( ◡ E ↾ 𝐴 )
2		rncnvepres	⊢ ran ( ◡ E ↾ 𝐴 ) = ∪ 𝐴
3	1 2	eqtri	⊢ dom ≀ ( ◡ E ↾ 𝐴 ) = ∪ 𝐴