Metamath Proof Explorer

Theorem bnj1052

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 bnj1052.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj1052.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj1052.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1052.4 
|- ( th <-> ( R _FrSe A /\ X e. A ) )
bnj1052.5 
|- ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
bnj1052.6 
|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
bnj1052.7 
|- D = ( _om \ { (/) } )
bnj1052.8 
|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1052.9 
|- ( et <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
bnj1052.10 
|- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
bnj1052.37 
|- ( ( th /\ ta /\ ch /\ ze ) -> ( _E Fr n /\ A. i e. n ( rh -> et ) ) )
Assertion bnj1052 
|- ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B )

Proof

Step Hyp Ref Expression
1 bnj1052.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj1052.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj1052.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj1052.4 
 |-  ( th <-> ( R _FrSe A /\ X e. A ) )
5 bnj1052.5 
 |-  ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
6 bnj1052.6 
 |-  ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
7 bnj1052.7 
 |-  D = ( _om \ { (/) } )
8 bnj1052.8 
 |-  K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9 bnj1052.9 
 |-  ( et <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
10 bnj1052.10 
 |-  ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
11 bnj1052.37 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> ( _E Fr n /\ A. i e. n ( rh -> et ) ) )
12 19.23vv 
 |-  ( A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> ( E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
13 12 albii 
 |-  ( A. f A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> A. f ( E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
14 19.23v 
 |-  ( A. f ( E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
15 13 14 bitri 
 |-  ( A. f A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
16 vex 
 |-  n e. _V
17 16 10 bnj110 
 |-  ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> A. i e. n et )
18 6 9 bnj1049 
 |-  ( A. i e. n et <-> A. i et )
19 17 18 sylib 
 |-  ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> A. i et )
20 19 19.21bi 
 |-  ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> et )
21 20 9 sylib 
 |-  ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
22 11 21 mpcom 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
23 22 gen2 
 |-  A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
24 15 23 mpgbi 
 |-  ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B )
25 1 2 3 4 5 6 7 8 24 bnj1034 
 |-  ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B )