Theorem dmcoels

Description: The domain of coelements in A is the union of A . (Contributed by Rodolfo Medina, 14-Oct-2010) (Revised by Peter Mazsa, 5-Apr-2018) (Revised by Peter Mazsa, 26-Sep-2021)

	Ref	Expression
Assertion	dmcoels	⊢ dom ∼ 𝐴 = ∪ 𝐴

Proof

Step	Hyp	Ref	Expression
1		df-coels	⊢ ∼ 𝐴 = ≀ ( ◡ E ↾ 𝐴 )
2	1	dmeqi	⊢ dom ∼ 𝐴 = dom ≀ ( ◡ E ↾ 𝐴 )
3		dm1cosscnvepres	⊢ dom ≀ ( ◡ E ↾ 𝐴 ) = ∪ 𝐴
4	2 3	eqtri	⊢ dom ∼ 𝐴 = ∪ 𝐴