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

Proof

Step	Hyp	Ref	Expression
1		epini.1	$$⊢A\in \mathrm{V}$$
2		vex	$$⊢x\in \mathrm{V}$$
3	2	eliniseg	$$⊢A\in \mathrm{V}\to \left(x\in {\mathrm{E}}^{-1}\left[\left\{A\right\}\right]\leftrightarrow x\mathrm{E}A\right)$$
4	1 3	ax-mp	$$⊢x\in {\mathrm{E}}^{-1}\left[\left\{A\right\}\right]\leftrightarrow x\mathrm{E}A$$
5	1	epeli	$$⊢x\mathrm{E}A\leftrightarrow x\in A$$
6	4 5	bitri	$$⊢x\in {\mathrm{E}}^{-1}\left[\left\{A\right\}\right]\leftrightarrow x\in A$$
7	6	eqriv	$$⊢{\mathrm{E}}^{-1}\left[\left\{A\right\}\right]=A$$