Theorem cnvepima

Description: The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023)

	Ref	Expression
Assertion	cnvepima	$$⊢A\in V\to {\mathrm{E}}^{-1}\left[A\right]=\bigcup A$$

Proof

Step	Hyp	Ref	Expression
1		cnvepresex	$$⊢A\in V\to {{\mathrm{E}}^{-1}\upharpoonright }_A\in \mathrm{V}$$
2		uniqs	$$⊢{{\mathrm{E}}^{-1}\upharpoonright }_A\in \mathrm{V}\to \bigcup \left(A/{\mathrm{E}}^{-1}\right)={\mathrm{E}}^{-1}\left[A\right]$$
3	1 2	syl	$$⊢A\in V\to \bigcup \left(A/{\mathrm{E}}^{-1}\right)={\mathrm{E}}^{-1}\left[A\right]$$
4		qsid	$$⊢A/{\mathrm{E}}^{-1}=A$$
5	4	unieqi	$$⊢\bigcup \left(A/{\mathrm{E}}^{-1}\right)=\bigcup A$$
6	3 5	eqtr3di	$$⊢A\in V\to {\mathrm{E}}^{-1}\left[A\right]=\bigcup A$$