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Metamath Proof Explorer

Theorem bnj998

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 bnj998.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj998.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj998.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj998.4 
|- ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
bnj998.5 
|- ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
bnj998.7 
|- ( ph' <-> [. p / n ]. ph )
bnj998.8 
|- ( ps' <-> [. p / n ]. ps )
bnj998.9 
|- ( ch' <-> [. p / n ]. ch )
bnj998.10 
|- ( ph" <-> [. G / f ]. ph' )
bnj998.11 
|- ( ps" <-> [. G / f ]. ps' )
bnj998.12 
|- ( ch" <-> [. G / f ]. ch' )
bnj998.13 
|- D = ( _om \ { (/) } )
bnj998.14 
|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj998.15 
|- C = U_ y e. ( f ` m ) _pred ( y , A , R )
bnj998.16 
|- G = ( f u. { <. n , C >. } )
Assertion bnj998 
|- ( ( th /\ ch /\ ta /\ et ) -> ch" )

Proof

Step Hyp Ref Expression
1 bnj998.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj998.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj998.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj998.4 
 |-  ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
5 bnj998.5 
 |-  ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
6 bnj998.7 
 |-  ( ph' <-> [. p / n ]. ph )
7 bnj998.8 
 |-  ( ps' <-> [. p / n ]. ps )
8 bnj998.9 
 |-  ( ch' <-> [. p / n ]. ch )
9 bnj998.10 
 |-  ( ph" <-> [. G / f ]. ph' )
10 bnj998.11 
 |-  ( ps" <-> [. G / f ]. ps' )
11 bnj998.12 
 |-  ( ch" <-> [. G / f ]. ch' )
12 bnj998.13 
 |-  D = ( _om \ { (/) } )
13 bnj998.14 
 |-  B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
14 bnj998.15 
 |-  C = U_ y e. ( f ` m ) _pred ( y , A , R )
15 bnj998.16 
 |-  G = ( f u. { <. n , C >. } )
16 bnj253 
 |-  ( ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) <-> ( ( R _FrSe A /\ X e. A ) /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
17 16 simp1bi 
 |-  ( ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) -> ( R _FrSe A /\ X e. A ) )
18 4 17 sylbi 
 |-  ( th -> ( R _FrSe A /\ X e. A ) )
19 18 bnj705 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( R _FrSe A /\ X e. A ) )
20 bnj643 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ch )
21 3simpc 
 |-  ( ( m e. _om /\ n = suc m /\ p = suc n ) -> ( n = suc m /\ p = suc n ) )
22 5 21 sylbi 
 |-  ( ta -> ( n = suc m /\ p = suc n ) )
23 22 bnj707 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( n = suc m /\ p = suc n ) )
24 bnj255 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ch /\ n = suc m /\ p = suc n ) <-> ( ( R _FrSe A /\ X e. A ) /\ ch /\ ( n = suc m /\ p = suc n ) ) )
25 19 20 23 24 syl3anbrc 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( ( R _FrSe A /\ X e. A ) /\ ch /\ n = suc m /\ p = suc n ) )
26 bnj252 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ch /\ n = suc m /\ p = suc n ) <-> ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) )
27 25 26 sylib 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) )
28 biid 
 |-  ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
29 biid 
 |-  ( ( n e. D /\ p = suc n /\ m e. n ) <-> ( n e. D /\ p = suc n /\ m e. n ) )
30 1 2 3 6 7 8 9 10 11 12 13 14 15 28 29 bnj910 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ch" )
31 27 30 syl 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ch" )