Theorem epini

Description: Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013)

	Ref	Expression
Hypothesis	epini.1	⊢ 𝐴 ∈ V
Assertion	epini	⊢ ( ◡ E “ { 𝐴 } ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		epini.1	⊢ 𝐴 ∈ V
2		vex	⊢ 𝑥 ∈ V
3	2	eliniseg	⊢ ( 𝐴 ∈ V → ( 𝑥 ∈ ( ◡ E “ { 𝐴 } ) ↔ 𝑥 E 𝐴 ) )
4	1 3	ax-mp	⊢ ( 𝑥 ∈ ( ◡ E “ { 𝐴 } ) ↔ 𝑥 E 𝐴 )
5	1	epeli	⊢ ( 𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴 )
6	4 5	bitri	⊢ ( 𝑥 ∈ ( ◡ E “ { 𝐴 } ) ↔ 𝑥 ∈ 𝐴 )
7	6	eqriv	⊢ ( ◡ E “ { 𝐴 } ) = 𝐴