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Metamath Proof Explorer

Theorem bnj981

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 bnj981.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj981.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj981.3 
|- D = ( _om \ { (/) } )
bnj981.4 
|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj981.5 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
Assertion bnj981 
|- ( Z e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) )

Proof

Step Hyp Ref Expression
1 bnj981.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj981.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj981.3 
 |-  D = ( _om \ { (/) } )
4 bnj981.4 
 |-  B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5 bnj981.5 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
6 nfv 
 |-  F/ y Z e. _trCl ( X , A , R )
7 nfcv 
 |-  F/_ y _om
8 nfv 
 |-  F/ y suc i e. n
9 nfiu1 
 |-  F/_ y U_ y e. ( f ` i ) _pred ( y , A , R )
10 9 nfeq2 
 |-  F/ y ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R )
11 8 10 nfim 
 |-  F/ y ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) )
12 7 11 nfralw 
 |-  F/ y A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) )
13 2 12 nfxfr 
 |-  F/ y ps
14 13 nf5ri 
 |-  ( ps -> A. y ps )
15 14 5 bnj1096 
 |-  ( ch -> A. y ch )
16 15 nf5i 
 |-  F/ y ch
17 nfv 
 |-  F/ y i e. n
18 nfv 
 |-  F/ y Z e. ( f ` i )
19 16 17 18 nf3an 
 |-  F/ y ( ch /\ i e. n /\ Z e. ( f ` i ) )
20 19 nfex 
 |-  F/ y E. i ( ch /\ i e. n /\ Z e. ( f ` i ) )
21 20 nfex 
 |-  F/ y E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) )
22 21 nfex 
 |-  F/ y E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) )
23 6 22 nfim 
 |-  F/ y ( Z e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) )
24 eleq1 
 |-  ( y = Z -> ( y e. _trCl ( X , A , R ) <-> Z e. _trCl ( X , A , R ) ) )
25 eleq1 
 |-  ( y = Z -> ( y e. ( f ` i ) <-> Z e. ( f ` i ) ) )
26 25 3anbi3d 
 |-  ( y = Z -> ( ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> ( ch /\ i e. n /\ Z e. ( f ` i ) ) ) )
27 26 3exbidv 
 |-  ( y = Z -> ( E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) ) )
28 24 27 imbi12d 
 |-  ( y = Z -> ( ( y e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) ) <-> ( Z e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) ) ) )
29 1 2 3 4 5 bnj917 
 |-  ( y e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) )
30 23 28 29 vtoclg1f 
 |-  ( Z e. _trCl ( X , A , R ) -> ( Z e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) ) )
31 30 pm2.43i 
 |-  ( Z e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) )