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Metamath Proof Explorer

Theorem bnj600

Description: Technical lemma for bnj852 . 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 bnj600.1 
|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
bnj600.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj600.3 
|- D = ( _om \ { (/) } )
bnj600.4 
|- ( ch <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
bnj600.5 
|- ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
bnj600.10 
|- ( ph' <-> [. m / n ]. ph )
bnj600.11 
|- ( ps' <-> [. m / n ]. ps )
bnj600.12 
|- ( ch' <-> [. m / n ]. ch )
bnj600.13 
|- ( ph" <-> [. G / f ]. ph )
bnj600.14 
|- ( ps" <-> [. G / f ]. ps )
bnj600.15 
|- ( ch" <-> [. G / f ]. ch )
bnj600.16 
|- G = ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } )
bnj600.17 
|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj600.18 
|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj600.19 
|- ( et <-> ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
bnj600.20 
|- ( ze <-> ( i e. _om /\ suc i e. n /\ m = suc i ) )
bnj600.21 
|- ( rh <-> ( i e. _om /\ suc i e. n /\ m =/= suc i ) )
bnj600.22 
|- B = U_ y e. ( f ` i ) _pred ( y , A , R )
bnj600.23 
|- C = U_ y e. ( f ` p ) _pred ( y , A , R )
bnj600.24 
|- K = U_ y e. ( G ` i ) _pred ( y , A , R )
bnj600.25 
|- L = U_ y e. ( G ` p ) _pred ( y , A , R )
bnj600.26 
|- G = ( f u. { <. m , C >. } )
Assertion bnj600 
|- ( n =/= 1o -> ( ( n e. D /\ th ) -> ch ) )

Proof

Step Hyp Ref Expression
1 bnj600.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
2 bnj600.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj600.3 
 |-  D = ( _om \ { (/) } )
4 bnj600.4 
 |-  ( ch <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
5 bnj600.5 
 |-  ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
6 bnj600.10 
 |-  ( ph' <-> [. m / n ]. ph )
7 bnj600.11 
 |-  ( ps' <-> [. m / n ]. ps )
8 bnj600.12 
 |-  ( ch' <-> [. m / n ]. ch )
9 bnj600.13 
 |-  ( ph" <-> [. G / f ]. ph )
10 bnj600.14 
 |-  ( ps" <-> [. G / f ]. ps )
11 bnj600.15 
 |-  ( ch" <-> [. G / f ]. ch )
12 bnj600.16 
 |-  G = ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } )
13 bnj600.17 
 |-  ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
14 bnj600.18 
 |-  ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
15 bnj600.19 
 |-  ( et <-> ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
16 bnj600.20 
 |-  ( ze <-> ( i e. _om /\ suc i e. n /\ m = suc i ) )
17 bnj600.21 
 |-  ( rh <-> ( i e. _om /\ suc i e. n /\ m =/= suc i ) )
18 bnj600.22 
 |-  B = U_ y e. ( f ` i ) _pred ( y , A , R )
19 bnj600.23 
 |-  C = U_ y e. ( f ` p ) _pred ( y , A , R )
20 bnj600.24 
 |-  K = U_ y e. ( G ` i ) _pred ( y , A , R )
21 bnj600.25 
 |-  L = U_ y e. ( G ` p ) _pred ( y , A , R )
22 bnj600.26 
 |-  G = ( f u. { <. m , C >. } )
23 12 bnj528 
 |-  G e. _V
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 1 9 23 bnj609 
 |-  ( ph" <-> ( G ` (/) ) = _pred ( x , A , R ) )
27 2 10 23 bnj611 
 |-  ( ps" <-> A. i e. _om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) _pred ( y , A , R ) ) )
28 3 bnj168 
 |-  ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )
29 df-rex 
 |-  ( E. m e. D n = suc m <-> E. m ( m e. D /\ n = suc m ) )
30 28 29 sylib 
 |-  ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. D /\ n = suc m ) )
31 3 bnj158 
 |-  ( m e. D -> E. p e. _om m = suc p )
32 df-rex 
 |-  ( E. p e. _om m = suc p <-> E. p ( p e. _om /\ m = suc p ) )
33 31 32 sylib 
 |-  ( m e. D -> E. p ( p e. _om /\ m = suc p ) )
34 33 adantr 
 |-  ( ( m e. D /\ n = suc m ) -> E. p ( p e. _om /\ m = suc p ) )
35 34 ancri 
 |-  ( ( m e. D /\ n = suc m ) -> ( E. p ( p e. _om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
36 35 bnj534 
 |-  ( ( m e. D /\ n = suc m ) -> E. p ( ( p e. _om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
37 bnj432 
 |-  ( ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) <-> ( ( p e. _om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
38 37 exbii 
 |-  ( E. p ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) <-> E. p ( ( p e. _om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
39 36 38 sylibr 
 |-  ( ( m e. D /\ n = suc m ) -> E. p ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
40 39 eximi 
 |-  ( E. m ( m e. D /\ n = suc m ) -> E. m E. p ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
41 30 40 syl 
 |-  ( ( n =/= 1o /\ n e. D ) -> E. m E. p ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
42 15 2exbii 
 |-  ( E. m E. p et <-> E. m E. p ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
43 41 42 sylibr 
 |-  ( ( n =/= 1o /\ n e. D ) -> E. m E. p et )
44 rsp 
 |-  ( A. m e. D ( m _E n -> [. m / n ]. ch ) -> ( m e. D -> ( m _E n -> [. m / n ]. ch ) ) )
45 5 44 sylbi 
 |-  ( th -> ( m e. D -> ( m _E n -> [. m / n ]. ch ) ) )
46 45 3imp 
 |-  ( ( th /\ m e. D /\ m _E n ) -> [. m / n ]. ch )
47 46 8 sylibr 
 |-  ( ( th /\ m e. D /\ m _E n ) -> ch' )
48 1 6 24 bnj523 
 |-  ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) )
49 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 ) ) )
50 48 49 3 12 13 14 bnj544 
 |-  ( ( R _FrSe A /\ ta /\ si ) -> G Fn n )
51 14 15 50 bnj561 
 |-  ( ( R _FrSe A /\ ta /\ et ) -> G Fn n )
52 48 3 12 13 14 50 26 bnj545 
 |-  ( ( R _FrSe A /\ ta /\ si ) -> ph" )
53 14 15 52 bnj562 
 |-  ( ( R _FrSe A /\ ta /\ et ) -> ph" )
54 3 12 13 14 15 16 18 19 20 21 22 48 49 50 17 51 27 bnj571 
 |-  ( ( R _FrSe A /\ ta /\ et ) -> ps" )
55 biid 
 |-  ( [. z / f ]. ph <-> [. z / f ]. ph )
56 biid 
 |-  ( [. z / f ]. ps <-> [. z / f ]. ps )
57 biid 
 |-  ( [. G / z ]. [. z / f ]. ph <-> [. G / z ]. [. z / f ]. ph )
58 biid 
 |-  ( [. G / z ]. [. z / f ]. ps <-> [. G / z ]. [. z / f ]. ps )
59 5 9 10 13 15 23 25 26 27 43 47 51 53 54 1 2 55 56 57 58 bnj607 
 |-  ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R _FrSe A /\ x e. A ) -> E. f ( f Fn n /\ ph /\ ps ) ) )
60 1 2 3 bnj579 
 |-  ( n e. D -> E* f ( f Fn n /\ ph /\ ps ) )
61 60 a1d 
 |-  ( n e. D -> ( ( R _FrSe A /\ x e. A ) -> E* f ( f Fn n /\ ph /\ ps ) ) )
62 61 3ad2ant2 
 |-  ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R _FrSe A /\ x e. A ) -> E* f ( f Fn n /\ ph /\ ps ) ) )
63 59 62 jcad 
 |-  ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R _FrSe A /\ x e. A ) -> ( E. f ( f Fn n /\ ph /\ ps ) /\ E* f ( f Fn n /\ ph /\ ps ) ) ) )
64 df-eu 
 |-  ( E! f ( f Fn n /\ ph /\ ps ) <-> ( E. f ( f Fn n /\ ph /\ ps ) /\ E* f ( f Fn n /\ ph /\ ps ) ) )
65 63 64 imbitrrdi 
 |-  ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
66 65 4 sylibr 
 |-  ( ( n =/= 1o /\ n e. D /\ th ) -> ch )
67 66 3expib 
 |-  ( n =/= 1o -> ( ( n e. D /\ th ) -> ch ) )