Theorem elcnvcnvlem

Description: Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020)

	Ref	Expression
Assertion	elcnvcnvlem	$$⊢A\in {B^{-1}}^{-1}\leftrightarrow \left(A\in \left(\mathrm{V}\times \mathrm{V}\right)\wedge \mathrm{I}\left(A\right)\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		cnvcnv	$$⊢{B^{-1}}^{-1}=B\cap \left(\mathrm{V}\times \mathrm{V}\right)$$
2		incom	$$⊢B\cap \left(\mathrm{V}\times \mathrm{V}\right)=\left(\mathrm{V}\times \mathrm{V}\right)\cap B$$
3	1 2	eqtri	$$⊢{B^{-1}}^{-1}=\left(\mathrm{V}\times \mathrm{V}\right)\cap B$$
4	3	eleq2i	$$⊢A\in {B^{-1}}^{-1}\leftrightarrow A\in \left(\left(\mathrm{V}\times \mathrm{V}\right)\cap B\right)$$
5		elinlem	$$⊢A\in \left(\left(\mathrm{V}\times \mathrm{V}\right)\cap B\right)\leftrightarrow \left(A\in \left(\mathrm{V}\times \mathrm{V}\right)\wedge \mathrm{I}\left(A\right)\in B\right)$$
6	4 5	bitri	$$⊢A\in {B^{-1}}^{-1}\leftrightarrow \left(A\in \left(\mathrm{V}\times \mathrm{V}\right)\wedge \mathrm{I}\left(A\right)\in B\right)$$