Theorem bj-elid5

Description: Characterization of the elements of _I . (Contributed by BJ, 22-Jun-2019)

	Ref	Expression
Assertion	bj-elid5	|- ( A e. _I <-> ( A e. ( _V X. _V ) /\ ( 1st ` A ) = ( 2nd ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		reli	|- Rel _I
2		df-rel	|- ( Rel _I <-> _I C_ ( _V X. _V ) )
3	1 2	mpbi	|- _I C_ ( _V X. _V )
4	3	sseli	|- ( A e. _I -> A e. ( _V X. _V ) )
5		bj-elid4	|- ( A e. ( _V X. _V ) -> ( A e. _I <-> ( 1st ` A ) = ( 2nd ` A ) ) )
6	4 5	biadanii	|- ( A e. _I <-> ( A e. ( _V X. _V ) /\ ( 1st ` A ) = ( 2nd ` A ) ) )