Theorem bj-cnvcnv

Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003) (Proof shortened by BJ, 26-Nov-2021) (Proof shortened by BJ, 16-Oct-2026) Prefer use of bj-cnvcnv2 . (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-cnvcnv	$$⊢{A^{-1}}^{-1}=A\cap \left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-cnvcnv2	$$⊢{A^{-1}}^{-1}={A\upharpoonright }_{\mathrm{V}}$$
2		df-res	$$⊢{A\upharpoonright }_{\mathrm{V}}=A\cap \left(\mathrm{V}\times \mathrm{V}\right)$$
3	1 2	eqtri	$$⊢{A^{-1}}^{-1}=A\cap \left(\mathrm{V}\times \mathrm{V}\right)$$