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 e. _V
Assertion	epini	|- ( `' _E " { A } ) = A

Proof

Step	Hyp	Ref	Expression
1		epini.1	|- A e. _V
2		vex	|- x e. _V
3	2	eliniseg	|- ( A e. _V -> ( x e. ( `' _E " { A } ) <-> x _E A ) )
4	1 3	ax-mp	|- ( x e. ( `' _E " { A } ) <-> x _E A )
5	1	epeli	|- ( x _E A <-> x e. A )
6	4 5	bitri	|- ( x e. ( `' _E " { A } ) <-> x e. A )
7	6	eqriv	|- ( `' _E " { A } ) = A