Theorem bnj1112

Description: Technical lemma for bnj69 . 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	bnj1112.1	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
Assertion	bnj1112	|- ( ps <-> A. j ( ( j e. _om /\ suc j e. n ) -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1112.1	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
2	1	bnj115	|- ( ps <-> A. i ( ( i e. _om /\ suc i e. n ) -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3		eleq1w	|- ( i = j -> ( i e. _om <-> j e. _om ) )
4		suceq	|- ( i = j -> suc i = suc j )
5	4	eleq1d	|- ( i = j -> ( suc i e. n <-> suc j e. n ) )
6	3 5	anbi12d	|- ( i = j -> ( ( i e. _om /\ suc i e. n ) <-> ( j e. _om /\ suc j e. n ) ) )
7	4	fveq2d	|- ( i = j -> ( f ` suc i ) = ( f ` suc j ) )
8		fveq2	|- ( i = j -> ( f ` i ) = ( f ` j ) )
9	8	bnj1113	|- ( i = j -> U_ y e. ( f ` i ) _pred ( y , A , R ) = U_ y e. ( f ` j ) _pred ( y , A , R ) )
10	7 9	eqeq12d	|- ( i = j -> ( ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) <-> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) )
11	6 10	imbi12d	|- ( i = j -> ( ( ( i e. _om /\ suc i e. n ) -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) <-> ( ( j e. _om /\ suc j e. n ) -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) )
12	11	cbvalvw	|- ( A. i ( ( i e. _om /\ suc i e. n ) -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) <-> A. j ( ( j e. _om /\ suc j e. n ) -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) )
13	2 12	bitri	|- ( ps <-> A. j ( ( j e. _om /\ suc j e. n ) -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) )