Theorem erdm

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

	Ref	Expression
Assertion	erdm	$$⊢R\mathrm{Er}A\to \mathrm{dom}R=A$$

Proof

Step	Hyp	Ref	Expression
1		df-er	$$⊢R\mathrm{Er}A\leftrightarrow \left(\mathrm{Rel}R\wedge \mathrm{dom}R=A\wedge R^{-1}\cup \left(R\circ R\right)\subseteq R\right)$$
2	1	simp2bi	$$⊢R\mathrm{Er}A\to \mathrm{dom}R=A$$