Theorem bnj222

Description: Technical lemma for bnj229 . 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
Hypothesis	bnj222.1	|- ( ps <-> A. i e. _om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) _pred ( y , A , R ) ) )
Assertion	bnj222	|- ( ps <-> A. m e. _om ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj222.1	|- ( ps <-> A. i e. _om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) _pred ( y , A , R ) ) )
2		suceq	|- ( i = m -> suc i = suc m )
3	2	eleq1d	|- ( i = m -> ( suc i e. N <-> suc m e. N ) )
4	2	fveq2d	|- ( i = m -> ( F ` suc i ) = ( F ` suc m ) )
5		fveq2	|- ( i = m -> ( F ` i ) = ( F ` m ) )
6	5	bnj1113	|- ( i = m -> U_ y e. ( F ` i ) _pred ( y , A , R ) = U_ y e. ( F ` m ) _pred ( y , A , R ) )
7	4 6	eqeq12d	|- ( i = m -> ( ( F ` suc i ) = U_ y e. ( F ` i ) _pred ( y , A , R ) <-> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )
8	3 7	imbi12d	|- ( i = m -> ( ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) _pred ( y , A , R ) ) <-> ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) ) )
9	8	cbvralvw	|- ( A. i e. _om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) _pred ( y , A , R ) ) <-> A. m e. _om ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )
10	1 9	bitri	|- ( ps <-> A. m e. _om ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )