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Metamath Proof Explorer

Theorem epin

Description: Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013) Put in closed form. (Revised by BJ, 16-Oct-2024)

		Ref	Expression
	Assertion	epin	$⊢ A \in V \to E^{-1} [\{A\}] = A$

Proof

Step	Hyp	Ref	Expression
1		vex	$⊢ x \in V$
2	1	eliniseg	$⊢ A \in V \to (x \in E^{-1} [\{A\}] \leftrightarrow x E A)$
3		epelg	$⊢ A \in V \to (x E A \leftrightarrow x \in A)$
4	2 3	bitrd	$⊢ A \in V \to (x \in E^{-1} [\{A\}] \leftrightarrow x \in A)$
5	4	eqrdv	$⊢ A \in V \to E^{-1} [\{A\}] = A$