Metamath Proof Explorer

Theorem bnj910

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 bnj910.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj910.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj910.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj910.4 
|- ( ph' <-> [. p / n ]. ph )
bnj910.5 
|- ( ps' <-> [. p / n ]. ps )
bnj910.6 
|- ( ch' <-> [. p / n ]. ch )
bnj910.7 
|- ( ph" <-> [. G / f ]. ph' )
bnj910.8 
|- ( ps" <-> [. G / f ]. ps' )
bnj910.9 
|- ( ch" <-> [. G / f ]. ch' )
bnj910.10 
|- D = ( _om \ { (/) } )
bnj910.11 
|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj910.12 
|- C = U_ y e. ( f ` m ) _pred ( y , A , R )
bnj910.13 
|- G = ( f u. { <. n , C >. } )
bnj910.14 
|- ( ta <-> ( f Fn n /\ ph /\ ps ) )
bnj910.15 
|- ( si <-> ( n e. D /\ p = suc n /\ m e. n ) )
Assertion bnj910 
|- ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ch" )

Proof

Step Hyp Ref Expression
1 bnj910.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj910.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj910.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj910.4 
 |-  ( ph' <-> [. p / n ]. ph )
5 bnj910.5 
 |-  ( ps' <-> [. p / n ]. ps )
6 bnj910.6 
 |-  ( ch' <-> [. p / n ]. ch )
7 bnj910.7 
 |-  ( ph" <-> [. G / f ]. ph' )
8 bnj910.8 
 |-  ( ps" <-> [. G / f ]. ps' )
9 bnj910.9 
 |-  ( ch" <-> [. G / f ]. ch' )
10 bnj910.10 
 |-  D = ( _om \ { (/) } )
11 bnj910.11 
 |-  B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
12 bnj910.12 
 |-  C = U_ y e. ( f ` m ) _pred ( y , A , R )
13 bnj910.13 
 |-  G = ( f u. { <. n , C >. } )
14 bnj910.14 
 |-  ( ta <-> ( f Fn n /\ ph /\ ps ) )
15 bnj910.15 
 |-  ( si <-> ( n e. D /\ p = suc n /\ m e. n ) )
16 3 10 bnj970 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> p e. D )
17 1 2 3 10 12 14 15 bnj969 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
18 simpr3 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> p = suc n )
19 3 bnj1235 
 |-  ( ch -> f Fn n )
20 19 3ad2ant1 
 |-  ( ( ch /\ n = suc m /\ p = suc n ) -> f Fn n )
21 20 adantl 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> f Fn n )
22 13 bnj941 
 |-  ( C e. _V -> ( ( p = suc n /\ f Fn n ) -> G Fn p ) )
23 22 3impib 
 |-  ( ( C e. _V /\ p = suc n /\ f Fn n ) -> G Fn p )
24 17 18 21 23 syl3anc 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> G Fn p )
25 1 2 3 4 7 10 12 13 14 15 bnj944 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ph" )
26 2 3 10 12 13 17 bnj967 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. _om /\ suc i e. p /\ suc i e. n ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) _pred ( y , A , R ) )
27 3 10 12 13 17 24 bnj966 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. _om /\ suc i e. p /\ n = suc i ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) _pred ( y , A , R ) )
28 2 3 5 8 12 13 26 27 bnj964 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ps" )
29 16 24 25 28 bnj951 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
30 vex 
 |-  p e. _V
31 3 4 5 6 30 bnj919 
 |-  ( ch' <-> ( p e. D /\ f Fn p /\ ph' /\ ps' ) )
32 13 bnj918 
 |-  G e. _V
33 31 7 8 9 32 bnj976 
 |-  ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
34 29 33 sylibr 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ch" )