Metamath Proof Explorer

Theorem bnj1021

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 bnj1021.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj1021.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj1021.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1021.4 
|- ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
bnj1021.5 
|- ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
bnj1021.6 
|- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
bnj1021.13 
|- D = ( _om \ { (/) } )
bnj1021.14 
|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
Assertion bnj1021 
|- E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) )

Proof

Step Hyp Ref Expression
1 bnj1021.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj1021.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj1021.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj1021.4 
 |-  ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
5 bnj1021.5 
 |-  ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
6 bnj1021.6 
 |-  ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
7 bnj1021.13 
 |-  D = ( _om \ { (/) } )
8 bnj1021.14 
 |-  B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9 1 2 3 4 5 6 7 8 bnj996 
 |-  E. f E. n E. i E. m E. p ( th -> ( ch /\ ta /\ et ) )
10 anclb 
 |-  ( ( th -> ( ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ( ch /\ ta /\ et ) ) ) )
11 bnj252 
 |-  ( ( th /\ ch /\ ta /\ et ) <-> ( th /\ ( ch /\ ta /\ et ) ) )
12 11 imbi2i 
 |-  ( ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ( ch /\ ta /\ et ) ) ) )
13 10 12 bitr4i 
 |-  ( ( th -> ( ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ ta /\ et ) ) )
14 13 2exbii 
 |-  ( E. m E. p ( th -> ( ch /\ ta /\ et ) ) <-> E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) )
15 14 3exbii 
 |-  ( E. f E. n E. i E. m E. p ( th -> ( ch /\ ta /\ et ) ) <-> E. f E. n E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) )
16 9 15 mpbi 
 |-  E. f E. n E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) )
17 19.37v 
 |-  ( E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> E. p ( th /\ ch /\ ta /\ et ) ) )
18 bnj1019 
 |-  ( E. p ( th /\ ch /\ ta /\ et ) <-> ( th /\ ch /\ et /\ E. p ta ) )
19 18 imbi2i 
 |-  ( ( th -> E. p ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
20 17 19 bitri 
 |-  ( E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
21 20 2exbii 
 |-  ( E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
22 21 2exbii 
 |-  ( E. f E. n E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
23 16 22 mpbi 
 |-  E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) )