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Metamath Proof Explorer

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 ) ) )