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Metamath Proof Explorer

Theorem bnj155

Description: Technical lemma for bnj153 . 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
Hypotheses bnj155.1 
|- ( ps1 <-> [. g / f ]. ps' )
bnj155.2 
|- ( ps' <-> A. i e. _om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
Assertion bnj155 
|- ( ps1 <-> A. i e. _om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) _pred ( y , A , R ) ) )

Proof

Step Hyp Ref Expression
1 bnj155.1 
 |-  ( ps1 <-> [. g / f ]. ps' )
2 bnj155.2 
 |-  ( ps' <-> A. i e. _om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 2 sbcbii 
 |-  ( [. g / f ]. ps' <-> [. g / f ]. A. i e. _om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
4 vex 
 |-  g e. _V
5 fveq1 
 |-  ( f = g -> ( f ` suc i ) = ( g ` suc i ) )
6 fveq1 
 |-  ( f = g -> ( f ` i ) = ( g ` i ) )
7 6 iuneq1d 
 |-  ( f = g -> U_ y e. ( f ` i ) _pred ( y , A , R ) = U_ y e. ( g ` i ) _pred ( y , A , R ) )
8 5 7 eqeq12d 
 |-  ( f = g -> ( ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) <-> ( g ` suc i ) = U_ y e. ( g ` i ) _pred ( y , A , R ) ) )
9 8 imbi2d 
 |-  ( f = g -> ( ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) <-> ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) _pred ( y , A , R ) ) ) )
10 9 ralbidv 
 |-  ( f = g -> ( A. i e. _om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) <-> A. i e. _om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) _pred ( y , A , R ) ) ) )
11 4 10 sbcie 
 |-  ( [. g / f ]. A. i e. _om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) <-> A. i e. _om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) _pred ( y , A , R ) ) )
12 1 3 11 3bitri 
 |-  ( ps1 <-> A. i e. _om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) _pred ( y , A , R ) ) )