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Metamath Proof Explorer

Theorem bnj579

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
Hypotheses bnj579.1 
|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
bnj579.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj579.3 
|- D = ( _om \ { (/) } )
Assertion bnj579 
|- ( n e. D -> E* f ( f Fn n /\ ph /\ ps ) )

Proof

Step Hyp Ref Expression
1 bnj579.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
2 bnj579.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj579.3 
 |-  D = ( _om \ { (/) } )
4 biid 
 |-  ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
5 biid 
 |-  ( [. g / f ]. ph <-> [. g / f ]. ph )
6 biid 
 |-  ( [. g / f ]. ps <-> [. g / f ]. ps )
7 biid 
 |-  ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> [. g / f ]. ( f Fn n /\ ph /\ ps ) )
8 biid 
 |-  ( ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) )
9 biid 
 |-  ( A. k e. n ( k _E j -> [. k / j ]. ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) ) <-> A. k e. n ( k _E j -> [. k / j ]. ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) ) )
10 1 2 4 5 6 7 3 8 9 bnj580 
 |-  ( n e. D -> E* f ( f Fn n /\ ph /\ ps ) )