Theorem elpredgg

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011) Generalize to closed form. (Revised by BJ, 16-Oct-2024)

	Ref	Expression
Assertion	elpredgg	|- ( ( X e. V /\ Y e. W ) -> ( Y e. Pred ( R , A , X ) <-> ( Y e. A /\ Y R X ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-pred	|- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) )
2	1	elin2	|- ( Y e. Pred ( R , A , X ) <-> ( Y e. A /\ Y e. ( `' R " { X } ) ) )
3		elinisegg	|- ( ( X e. V /\ Y e. W ) -> ( Y e. ( `' R " { X } ) <-> Y R X ) )
4	3	anbi2d	|- ( ( X e. V /\ Y e. W ) -> ( ( Y e. A /\ Y e. ( `' R " { X } ) ) <-> ( Y e. A /\ Y R X ) ) )
5	2 4	syl5bb	|- ( ( X e. V /\ Y e. W ) -> ( Y e. Pred ( R , A , X ) <-> ( Y e. A /\ Y R X ) ) )