Theorem eltrpred

Description: A class is a transitive predecessor iff it is in some value of the underlying function. This theorem is not really meant to be used directly: instead refer to trpredpred and trpredmintr . (Contributed by Scott Fenton, 28-Apr-2012)

	Ref	Expression
Assertion	eltrpred	|- ( Y e. TrPred ( R , A , X ) <-> E. i e. _om Y e. ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) ` i ) )

Proof

Step	Hyp	Ref	Expression
1		dftrpred2	|- TrPred ( R , A , X ) = U_ i e. _om ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) ` i )
2	1	eleq2i	|- ( Y e. TrPred ( R , A , X ) <-> Y e. U_ i e. _om ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) ` i ) )
3		eliun	|- ( Y e. U_ i e. _om ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) ` i ) <-> E. i e. _om Y e. ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) ` i ) )
4	2 3	bitri	|- ( Y e. TrPred ( R , A , X ) <-> E. i e. _om Y e. ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) ` i ) )