Theorem erdm

Description: The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015)

	Ref	Expression
Assertion	erdm	⊢ ( 𝑅 Er 𝐴 → dom 𝑅 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-er	⊢ ( 𝑅 Er 𝐴 ↔ ( Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) )
2	1	simp2bi	⊢ ( 𝑅 Er 𝐴 → dom 𝑅 = 𝐴 )