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 ∈ 𝑉 → ( ◡ E “ { 𝐴 } ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2	1	eliniseg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ ( ◡ E “ { 𝐴 } ) ↔ 𝑥 E 𝐴 ) )
3		epelg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴 ) )
4	2 3	bitrd	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ ( ◡ E “ { 𝐴 } ) ↔ 𝑥 ∈ 𝐴 ) )
5	4	eqrdv	⊢ ( 𝐴 ∈ 𝑉 → ( ◡ E “ { 𝐴 } ) = 𝐴 )