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	$$⊢A\in V\to {\mathrm{E}}^{-1}\left[\left\{A\right\}\right]=A$$

Proof

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