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Metamath Proof Explorer

Theorem bnj917

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
Hypotheses bnj917.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj917.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj917.3 
|- D = ( _om \ { (/) } )
bnj917.4 
|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj917.5 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
Assertion bnj917 
|- ( y e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) )

Proof

Step Hyp Ref Expression
1 bnj917.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj917.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj917.3 
 |-  D = ( _om \ { (/) } )
4 bnj917.4 
 |-  B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5 bnj917.5 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
6 biid 
 |-  ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
7 1 2 3 4 6 bnj916 
 |-  ( y e. _trCl ( X , A , R ) -> E. f E. n E. i ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
8 bnj252 
 |-  ( ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
9 5 8 bitri 
 |-  ( ch <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
10 9 3anbi1i 
 |-  ( ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) /\ i e. n /\ y e. ( f ` i ) ) )
11 bnj253 
 |-  ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) /\ i e. n /\ y e. ( f ` i ) ) )
12 10 11 bitr4i 
 |-  ( ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
13 12 3exbii 
 |-  ( E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> E. f E. n E. i ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
14 7 13 sylibr 
 |-  ( y e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) )