Metamath Proof Explorer

Theorem bnj590

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
Hypothesis bnj590.1 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
Assertion bnj590 
|- ( ( B = suc i /\ ps ) -> ( i e. _om -> ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) )

Proof

Step Hyp Ref Expression
1 bnj590.1 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
2 rsp 
 |-  ( A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) -> ( i e. _om -> ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) )
3 1 2 sylbi 
 |-  ( ps -> ( i e. _om -> ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) )
4 eleq1 
 |-  ( B = suc i -> ( B e. n <-> suc i e. n ) )
5 fveqeq2 
 |-  ( B = suc i -> ( ( f ` B ) = U_ y e. ( f ` i ) _pred ( y , A , R ) <-> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
6 4 5 imbi12d 
 |-  ( B = suc i -> ( ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) <-> ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) )
7 6 imbi2d 
 |-  ( B = suc i -> ( ( i e. _om -> ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) <-> ( i e. _om -> ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) ) )
8 3 7 syl5ibr 
 |-  ( B = suc i -> ( ps -> ( i e. _om -> ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) ) )
9 8 imp 
 |-  ( ( B = suc i /\ ps ) -> ( i e. _om -> ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) )