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Metamath Proof Explorer

Theorem elcnv

Description: Membership in a converse relation. Equation 5 of Suppes p. 62. (Contributed by NM, 24-Mar-1998)

		Ref	Expression
	Assertion	elcnv	$⊢ A \in R^{-1} \leftrightarrow \exists x \exists y (A = ⟨x, y⟩ \land y R x)$

Step	Hyp	Ref	Expression
1		df-cnv	$⊢ R^{-1} = \{⟨x, y⟩ \| y R x\}$
2	1	eleq2i	$⊢ A \in R^{-1} \leftrightarrow A \in \{⟨x, y⟩ \| y R x\}$
3		elopab	$⊢ A \in \{⟨x, y⟩ \| y R x\} \leftrightarrow \exists x \exists y (A = ⟨x, y⟩ \land y R x)$
4	2 3	bitri	$⊢ A \in R^{-1} \leftrightarrow \exists x \exists y (A = ⟨x, y⟩ \land y R x)$