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 = ( 𝐴 ∩ ( V × V ) )

Proof

Step	Hyp	Ref	Expression
1		bj-cnvcnv2	⊢ ◡ ◡ 𝐴 = ( 𝐴 ↾ V )
2		df-res	⊢ ( 𝐴 ↾ V ) = ( 𝐴 ∩ ( V × V ) )
3	1 2	eqtri	⊢ ◡ ◡ 𝐴 = ( 𝐴 ∩ ( V × V ) )