Theorem bnj965

Description: Technical lemma for bnj852 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj965.1	|- ( ps <-> A. i e. _om ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
	bnj965.2	|- ( ps" <-> [. G / f ]. ps )
	bnj965.12000	|- C = U_ y e. ( f ` m ) _pred ( y , A , R )
	bnj965.13000	|- G = ( f u. { <. n , C >. } )
Assertion	bnj965	|- ( ps" <-> A. i e. _om ( suc i e. N -> ( G ` suc i ) = U_ y e. ( G ` i ) _pred ( y , A , R ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj965.1	|- ( ps <-> A. i e. _om ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
2		bnj965.2	|- ( ps" <-> [. G / f ]. ps )
3		bnj965.12000	|- C = U_ y e. ( f ` m ) _pred ( y , A , R )
4		bnj965.13000	|- G = ( f u. { <. n , C >. } )
5	4	bnj918	|- G e. _V
6	1 2 5 3 4	bnj1000	|- ( ps" <-> A. i e. _om ( suc i e. N -> ( G ` suc i ) = U_ y e. ( G ` i ) _pred ( y , A , R ) ) )