Theorem predbrg

Description: Closed form of elpredim . (Contributed by Scott Fenton, 13-Apr-2011) (Revised by NM, 5-Apr-2016)

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

Proof

Step	Hyp	Ref	Expression
1		predeq3	|- ( x = X -> Pred ( R , A , x ) = Pred ( R , A , X ) )
2	1	eleq2d	|- ( x = X -> ( Y e. Pred ( R , A , x ) <-> Y e. Pred ( R , A , X ) ) )
3		breq2	|- ( x = X -> ( Y R x <-> Y R X ) )
4	2 3	imbi12d	|- ( x = X -> ( ( Y e. Pred ( R , A , x ) -> Y R x ) <-> ( Y e. Pred ( R , A , X ) -> Y R X ) ) )
5		vex	|- x e. _V
6	5	elpredim	|- ( Y e. Pred ( R , A , x ) -> Y R x )
7	4 6	vtoclg	|- ( X e. V -> ( Y e. Pred ( R , A , X ) -> Y R X ) )
8	7	imp	|- ( ( X e. V /\ Y e. Pred ( R , A , X ) ) -> Y R X )