Theorem elidinxpid

Description: Characterization of the elements of the intersection of the identity relation with a Cartesian square. (Contributed by Peter Mazsa, 9-Sep-2022)

	Ref	Expression
Assertion	elidinxpid	|- ( B e. ( _I i^i ( A X. A ) ) <-> E. x e. A B = <. x , x >. )

Proof

Step	Hyp	Ref	Expression
1		elidinxp	|- ( B e. ( _I i^i ( A X. A ) ) <-> E. x e. ( A i^i A ) B = <. x , x >. )
2		inidm	|- ( A i^i A ) = A
3	2	rexeqi	|- ( E. x e. ( A i^i A ) B = <. x , x >. <-> E. x e. A B = <. x , x >. )
4	1 3	bitri	|- ( B e. ( _I i^i ( A X. A ) ) <-> E. x e. A B = <. x , x >. )