Theorem dftrpred2

Description: A definition of the transitive predecessors of a class in terms of indexed union. (Contributed by Scott Fenton, 28-Apr-2012)

	Ref	Expression
Assertion	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 )

Proof

Step	Hyp	Ref	Expression
1		df-trpred	|- TrPred ( R , A , X ) = U. ran ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om )
2		frfnom	|- ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) Fn _om
3		fniunfv	|- ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) Fn _om -> U_ i e. _om ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) ` i ) = U. ran ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) )
4	2 3	ax-mp	|- U_ i e. _om ( ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om ) ` i ) = U. ran ( rec ( ( a e. _V |-> U_ y e. a Pred ( R , A , y ) ) , Pred ( R , A , X ) ) |` _om )
5	1 4	eqtr4i	|- 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 )