Metamath Proof Explorer

Theorem bnj907

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 bnj907.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj907.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj907.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj907.4 
|- ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
bnj907.5 
|- ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
bnj907.6 
|- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
bnj907.7 
|- ( ph' <-> [. p / n ]. ph )
bnj907.8 
|- ( ps' <-> [. p / n ]. ps )
bnj907.9 
|- ( ch' <-> [. p / n ]. ch )
bnj907.10 
|- ( ph" <-> [. G / f ]. ph' )
bnj907.11 
|- ( ps" <-> [. G / f ]. ps' )
bnj907.12 
|- ( ch" <-> [. G / f ]. ch' )
bnj907.13 
|- D = ( _om \ { (/) } )
bnj907.14 
|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj907.15 
|- C = U_ y e. ( f ` m ) _pred ( y , A , R )
bnj907.16 
|- G = ( f u. { <. n , C >. } )
Assertion bnj907 
|- ( ( R _FrSe A /\ X e. A ) -> _TrFo ( _trCl ( X , A , R ) , A , R ) )

Proof

Step Hyp Ref Expression
1 bnj907.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj907.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj907.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj907.4 
 |-  ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
5 bnj907.5 
 |-  ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
6 bnj907.6 
 |-  ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
7 bnj907.7 
 |-  ( ph' <-> [. p / n ]. ph )
8 bnj907.8 
 |-  ( ps' <-> [. p / n ]. ps )
9 bnj907.9 
 |-  ( ch' <-> [. p / n ]. ch )
10 bnj907.10 
 |-  ( ph" <-> [. G / f ]. ph' )
11 bnj907.11 
 |-  ( ps" <-> [. G / f ]. ps' )
12 bnj907.12 
 |-  ( ch" <-> [. G / f ]. ch' )
13 bnj907.13 
 |-  D = ( _om \ { (/) } )
14 bnj907.14 
 |-  B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
15 bnj907.15 
 |-  C = U_ y e. ( f ` m ) _pred ( y , A , R )
16 bnj907.16 
 |-  G = ( f u. { <. n , C >. } )
17 1 2 3 4 5 6 13 14 bnj1021 
 |-  E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) )
18 vex 
 |-  p e. _V
19 3 7 8 9 18 bnj919 
 |-  ( ch' <-> ( p e. D /\ f Fn p /\ ph' /\ ps' ) )
20 16 bnj918 
 |-  G e. _V
21 19 10 11 12 20 bnj976 
 |-  ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 21 bnj1020 
 |-  ( ( th /\ ch /\ et /\ E. p ta ) -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) )
23 22 ax-gen 
 |-  A. m ( ( th /\ ch /\ et /\ E. p ta ) -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) )
24 19.29r 
 |-  ( ( E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) /\ A. m ( ( th /\ ch /\ et /\ E. p ta ) -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) ) -> E. m ( ( th -> ( th /\ ch /\ et /\ E. p ta ) ) /\ ( ( th /\ ch /\ et /\ E. p ta ) -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) ) )
25 pm3.33 
 |-  ( ( ( th -> ( th /\ ch /\ et /\ E. p ta ) ) /\ ( ( th /\ ch /\ et /\ E. p ta ) -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) ) -> ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) )
26 24 25 bnj593 
 |-  ( ( E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) /\ A. m ( ( th /\ ch /\ et /\ E. p ta ) -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) ) -> E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) )
27 23 26 mpan2 
 |-  ( E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) -> E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) )
28 27 2eximi 
 |-  ( E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) -> E. n E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) )
29 17 28 bnj101 
 |-  E. f E. n E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) )
30 19.9v 
 |-  ( E. f E. n E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) <-> E. n E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) )
31 29 30 mpbi 
 |-  E. n E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) )
32 19.9v 
 |-  ( E. n E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) <-> E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) )
33 31 32 mpbi 
 |-  E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) )
34 19.9v 
 |-  ( E. i E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) <-> E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) )
35 33 34 mpbi 
 |-  E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) )
36 19.9v 
 |-  ( E. m ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) <-> ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) ) )
37 35 36 mpbi 
 |-  ( th -> _pred ( y , A , R ) C_ _trCl ( X , A , R ) )
38 4 bnj1254 
 |-  ( th -> z e. _pred ( y , A , R ) )
39 37 38 sseldd 
 |-  ( th -> z e. _trCl ( X , A , R ) )
40 4 39 bnj978 
 |-  ( ( R _FrSe A /\ X e. A ) -> _TrFo ( _trCl ( X , A , R ) , A , R ) )