Metamath Proof Explorer

Theorem bnj908

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 bnj908.1 
|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
bnj908.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj908.3 
|- D = ( _om \ { (/) } )
bnj908.4 
|- ( ch <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
bnj908.5 
|- ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
bnj908.10 
|- ( ph' <-> [. m / n ]. ph )
bnj908.11 
|- ( ps' <-> [. m / n ]. ps )
bnj908.12 
|- ( ch' <-> [. m / n ]. ch )
bnj908.13 
|- ( ph" <-> [. G / f ]. ph )
bnj908.14 
|- ( ps" <-> [. G / f ]. ps )
bnj908.15 
|- ( ch" <-> [. G / f ]. ch )
bnj908.16 
|- G = ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } )
bnj908.17 
|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj908.18 
|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj908.19 
|- ( et <-> ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
bnj908.20 
|- ( ze <-> ( i e. _om /\ suc i e. n /\ m = suc i ) )
bnj908.21 
|- ( rh <-> ( i e. _om /\ suc i e. n /\ m =/= suc i ) )
bnj908.22 
|- B = U_ y e. ( f ` i ) _pred ( y , A , R )
bnj908.23 
|- C = U_ y e. ( f ` p ) _pred ( y , A , R )
bnj908.24 
|- K = U_ y e. ( G ` i ) _pred ( y , A , R )
bnj908.25 
|- L = U_ y e. ( G ` p ) _pred ( y , A , R )
bnj908.26 
|- G = ( f u. { <. m , C >. } )
Assertion bnj908 
|- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) )

Proof

Step Hyp Ref Expression
1 bnj908.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
2 bnj908.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj908.3 
 |-  D = ( _om \ { (/) } )
4 bnj908.4 
 |-  ( ch <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
5 bnj908.5 
 |-  ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
6 bnj908.10 
 |-  ( ph' <-> [. m / n ]. ph )
7 bnj908.11 
 |-  ( ps' <-> [. m / n ]. ps )
8 bnj908.12 
 |-  ( ch' <-> [. m / n ]. ch )
9 bnj908.13 
 |-  ( ph" <-> [. G / f ]. ph )
10 bnj908.14 
 |-  ( ps" <-> [. G / f ]. ps )
11 bnj908.15 
 |-  ( ch" <-> [. G / f ]. ch )
12 bnj908.16 
 |-  G = ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } )
13 bnj908.17 
 |-  ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
14 bnj908.18 
 |-  ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
15 bnj908.19 
 |-  ( et <-> ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
16 bnj908.20 
 |-  ( ze <-> ( i e. _om /\ suc i e. n /\ m = suc i ) )
17 bnj908.21 
 |-  ( rh <-> ( i e. _om /\ suc i e. n /\ m =/= suc i ) )
18 bnj908.22 
 |-  B = U_ y e. ( f ` i ) _pred ( y , A , R )
19 bnj908.23 
 |-  C = U_ y e. ( f ` p ) _pred ( y , A , R )
20 bnj908.24 
 |-  K = U_ y e. ( G ` i ) _pred ( y , A , R )
21 bnj908.25 
 |-  L = U_ y e. ( G ` p ) _pred ( y , A , R )
22 bnj908.26 
 |-  G = ( f u. { <. m , C >. } )
23 bnj248 
 |-  ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) <-> ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) /\ et ) )
24 vex 
 |-  m e. _V
25 4 6 7 8 24 bnj207 
 |-  ( ch' <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
26 25 biimpi 
 |-  ( ch' -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
27 euex 
 |-  ( E! f ( f Fn m /\ ph' /\ ps' ) -> E. f ( f Fn m /\ ph' /\ ps' ) )
28 26 27 syl6 
 |-  ( ch' -> ( ( R _FrSe A /\ x e. A ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) )
29 28 impcom 
 |-  ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) -> E. f ( f Fn m /\ ph' /\ ps' ) )
30 29 13 bnj1198 
 |-  ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) -> E. f ta )
31 23 30 bnj832 
 |-  ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ta )
32 bnj645 
 |-  ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> et )
33 19.41v 
 |-  ( E. f ( ta /\ et ) <-> ( E. f ta /\ et ) )
34 31 32 33 sylanbrc 
 |-  ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ta /\ et ) )
35 bnj642 
 |-  ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> R _FrSe A )
36 19.41v 
 |-  ( E. f ( ( ta /\ et ) /\ R _FrSe A ) <-> ( E. f ( ta /\ et ) /\ R _FrSe A ) )
37 34 35 36 sylanbrc 
 |-  ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ( ta /\ et ) /\ R _FrSe A ) )
38 bnj170 
 |-  ( ( R _FrSe A /\ ta /\ et ) <-> ( ( ta /\ et ) /\ R _FrSe A ) )
39 37 38 bnj1198 
 |-  ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( R _FrSe A /\ ta /\ et ) )
40 1 6 24 bnj523 
 |-  ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) )
41 2 7 24 bnj539 
 |-  ( ps' <-> A. i e. _om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
42 40 41 3 12 13 14 bnj544 
 |-  ( ( R _FrSe A /\ ta /\ si ) -> G Fn n )
43 14 15 42 bnj561 
 |-  ( ( R _FrSe A /\ ta /\ et ) -> G Fn n )
44 12 bnj528 
 |-  G e. _V
45 1 9 44 bnj609 
 |-  ( ph" <-> ( G ` (/) ) = _pred ( x , A , R ) )
46 40 3 12 13 14 42 45 bnj545 
 |-  ( ( R _FrSe A /\ ta /\ si ) -> ph" )
47 14 15 46 bnj562 
 |-  ( ( R _FrSe A /\ ta /\ et ) -> ph" )
48 2 10 44 bnj611 
 |-  ( ps" <-> A. i e. _om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) _pred ( y , A , R ) ) )
49 3 12 13 14 15 16 18 19 20 21 22 40 41 42 17 43 48 bnj571 
 |-  ( ( R _FrSe A /\ ta /\ et ) -> ps" )
50 43 47 49 3jca 
 |-  ( ( R _FrSe A /\ ta /\ et ) -> ( G Fn n /\ ph" /\ ps" ) )
51 39 50 bnj593 
 |-  ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) )