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Metamath Proof Explorer

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 ) ) )