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 ∈ ◡ ◡ 𝐵 ↔ ( 𝐴 ∈ ( V × V ) ∧ ( I ‘ 𝐴 ) ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		cnvcnv	⊢ ◡ ◡ 𝐵 = ( 𝐵 ∩ ( V × V ) )
2		incom	⊢ ( 𝐵 ∩ ( V × V ) ) = ( ( V × V ) ∩ 𝐵 )
3	1 2	eqtri	⊢ ◡ ◡ 𝐵 = ( ( V × V ) ∩ 𝐵 )
4	3	eleq2i	⊢ ( 𝐴 ∈ ◡ ◡ 𝐵 ↔ 𝐴 ∈ ( ( V × V ) ∩ 𝐵 ) )
5		elinlem	⊢ ( 𝐴 ∈ ( ( V × V ) ∩ 𝐵 ) ↔ ( 𝐴 ∈ ( V × V ) ∧ ( I ‘ 𝐴 ) ∈ 𝐵 ) )
6	4 5	bitri	⊢ ( 𝐴 ∈ ◡ ◡ 𝐵 ↔ ( 𝐴 ∈ ( V × V ) ∧ ( I ‘ 𝐴 ) ∈ 𝐵 ) )