Theorem bj-eldiag

Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 . (Contributed by BJ, 22-Jun-2019)

	Ref	Expression
Assertion	bj-eldiag	|- ( A e. V -> ( B e. ( _Id ` A ) <-> ( B e. ( A X. A ) /\ ( 1st ` B ) = ( 2nd ` B ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-diagval2	|- ( A e. V -> ( _Id ` A ) = ( _I i^i ( A X. A ) ) )
2	1	eleq2d	|- ( A e. V -> ( B e. ( _Id ` A ) <-> B e. ( _I i^i ( A X. A ) ) ) )
3		elin	|- ( B e. ( _I i^i ( A X. A ) ) <-> ( B e. _I /\ B e. ( A X. A ) ) )
4		ancom	|- ( ( B e. _I /\ B e. ( A X. A ) ) <-> ( B e. ( A X. A ) /\ B e. _I ) )
5		bj-elid4	|- ( B e. ( A X. A ) -> ( B e. _I <-> ( 1st ` B ) = ( 2nd ` B ) ) )
6	5	pm5.32i	|- ( ( B e. ( A X. A ) /\ B e. _I ) <-> ( B e. ( A X. A ) /\ ( 1st ` B ) = ( 2nd ` B ) ) )
7	3 4 6	3bitri	|- ( B e. ( _I i^i ( A X. A ) ) <-> ( B e. ( A X. A ) /\ ( 1st ` B ) = ( 2nd ` B ) ) )
8	2 7	bitrdi	|- ( A e. V -> ( B e. ( _Id ` A ) <-> ( B e. ( A X. A ) /\ ( 1st ` B ) = ( 2nd ` B ) ) ) )