Metamath Proof Explorer

Theorem bnj1030

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 bnj1030.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj1030.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj1030.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1030.4 
|- ( th <-> ( R _FrSe A /\ X e. A ) )
bnj1030.5 
|- ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
bnj1030.6 
|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
bnj1030.7 
|- D = ( _om \ { (/) } )
bnj1030.8 
|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1030.9 
|- ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
bnj1030.10 
|- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
bnj1030.11 
|- ( ph' <-> [. j / i ]. ph )
bnj1030.12 
|- ( ps' <-> [. j / i ]. ps )
bnj1030.13 
|- ( ch' <-> [. j / i ]. ch )
bnj1030.14 
|- ( th' <-> [. j / i ]. th )
bnj1030.15 
|- ( ta' <-> [. j / i ]. ta )
bnj1030.16 
|- ( ze' <-> [. j / i ]. ze )
bnj1030.17 
|- ( et' <-> [. j / i ]. et )
bnj1030.18 
|- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) )
bnj1030.19 
|- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) )
Assertion bnj1030 
|- ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B )

Proof

Step Hyp Ref Expression
1 bnj1030.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj1030.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj1030.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj1030.4 
 |-  ( th <-> ( R _FrSe A /\ X e. A ) )
5 bnj1030.5 
 |-  ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
6 bnj1030.6 
 |-  ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
7 bnj1030.7 
 |-  D = ( _om \ { (/) } )
8 bnj1030.8 
 |-  K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9 bnj1030.9 
 |-  ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
10 bnj1030.10 
 |-  ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
11 bnj1030.11 
 |-  ( ph' <-> [. j / i ]. ph )
12 bnj1030.12 
 |-  ( ps' <-> [. j / i ]. ps )
13 bnj1030.13 
 |-  ( ch' <-> [. j / i ]. ch )
14 bnj1030.14 
 |-  ( th' <-> [. j / i ]. th )
15 bnj1030.15 
 |-  ( ta' <-> [. j / i ]. ta )
16 bnj1030.16 
 |-  ( ze' <-> [. j / i ]. ze )
17 bnj1030.17 
 |-  ( et' <-> [. j / i ]. et )
18 bnj1030.18 
 |-  ( si <-> ( ( j e. n /\ j _E i ) -> et' ) )
19 bnj1030.19 
 |-  ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) )
20 19.23vv 
 |-  ( A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) <-> ( E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
21 20 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 ) )
22 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 ) )
23 21 22 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 ) )
24 7 bnj1071 
 |-  ( n e. D -> _E Fr n )
25 3 24 bnj769 
 |-  ( ch -> _E Fr n )
26 25 bnj707 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> _E Fr n )
27 2 8 9 17 bnj1123 
 |-  ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )
28 2 3 5 7 18 19 27 bnj1118 
 |-  E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B )
29 1 3 5 bnj1097 
 |-  ( ( i = (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B )
30 28 29 bnj1109 
 |-  E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B )
31 30 2 3 bnj1093 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) )
32 9 10 17 18 19 31 bnj1090 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n ( rh -> et ) )
33 vex 
 |-  n e. _V
34 33 10 bnj110 
 |-  ( ( _E Fr n /\ A. i e. n ( rh -> et ) ) -> A. i e. n et )
35 26 32 34 syl2anc 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n et )
36 4 5 3 6 9 35 8 bnj1121 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
37 36 gen2 
 |-  A. n A. i ( ( th /\ ta /\ ch /\ ze ) -> z e. B )
38 23 37 mpgbi 
 |-  ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B )
39 1 2 3 4 5 6 7 8 38 bnj1034 
 |-  ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B )