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 e. `' `' B <-> ( A e. ( _V X. _V ) /\ ( _I ` A ) e. B ) )

Proof

Step	Hyp	Ref	Expression
1		cnvcnv	|- `' `' B = ( B i^i ( _V X. _V ) )
2		incom	|- ( B i^i ( _V X. _V ) ) = ( ( _V X. _V ) i^i B )
3	1 2	eqtri	|- `' `' B = ( ( _V X. _V ) i^i B )
4	3	eleq2i	|- ( A e. `' `' B <-> A e. ( ( _V X. _V ) i^i B ) )
5		elinlem	|- ( A e. ( ( _V X. _V ) i^i B ) <-> ( A e. ( _V X. _V ) /\ ( _I ` A ) e. B ) )
6	4 5	bitri	|- ( A e. `' `' B <-> ( A e. ( _V X. _V ) /\ ( _I ` A ) e. B ) )