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Metamath Proof Explorer

Theorem bnj1006

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 bnj1006.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj1006.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj1006.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1006.4 
|- ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
bnj1006.5 
|- ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
bnj1006.6 
|- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
bnj1006.7 
|- ( ph' <-> [. p / n ]. ph )
bnj1006.8 
|- ( ps' <-> [. p / n ]. ps )
bnj1006.9 
|- ( ch' <-> [. p / n ]. ch )
bnj1006.10 
|- ( ph" <-> [. G / f ]. ph' )
bnj1006.11 
|- ( ps" <-> [. G / f ]. ps' )
bnj1006.12 
|- ( ch" <-> [. G / f ]. ch' )
bnj1006.13 
|- D = ( _om \ { (/) } )
bnj1006.15 
|- C = U_ y e. ( f ` m ) _pred ( y , A , R )
bnj1006.16 
|- G = ( f u. { <. n , C >. } )
bnj1006.28 
|- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. _om /\ suc i e. p ) )
Assertion bnj1006 
|- ( ( th /\ ch /\ ta /\ et ) -> _pred ( y , A , R ) C_ ( G ` suc i ) )

Proof

Step Hyp Ref Expression
1 bnj1006.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj1006.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj1006.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj1006.4 
 |-  ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
5 bnj1006.5 
 |-  ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
6 bnj1006.6 
 |-  ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
7 bnj1006.7 
 |-  ( ph' <-> [. p / n ]. ph )
8 bnj1006.8 
 |-  ( ps' <-> [. p / n ]. ps )
9 bnj1006.9 
 |-  ( ch' <-> [. p / n ]. ch )
10 bnj1006.10 
 |-  ( ph" <-> [. G / f ]. ph' )
11 bnj1006.11 
 |-  ( ps" <-> [. G / f ]. ps' )
12 bnj1006.12 
 |-  ( ch" <-> [. G / f ]. ch' )
13 bnj1006.13 
 |-  D = ( _om \ { (/) } )
14 bnj1006.15 
 |-  C = U_ y e. ( f ` m ) _pred ( y , A , R )
15 bnj1006.16 
 |-  G = ( f u. { <. n , C >. } )
16 bnj1006.28 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. _om /\ suc i e. p ) )
17 6 simprbi 
 |-  ( et -> y e. ( f ` i ) )
18 17 bnj708 
 |-  ( ( th /\ ch /\ ta /\ et ) -> y e. ( f ` i ) )
19 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 ) ) )
20 19 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 ) )
21 4 20 sylbi 
 |-  ( th -> ( R _FrSe A /\ X e. A ) )
22 21 bnj705 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( R _FrSe A /\ X e. A ) )
23 bnj643 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ch )
24 3simpc 
 |-  ( ( m e. _om /\ n = suc m /\ p = suc n ) -> ( n = suc m /\ p = suc n ) )
25 5 24 sylbi 
 |-  ( ta -> ( n = suc m /\ p = suc n ) )
26 25 bnj707 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( n = suc m /\ p = suc n ) )
27 3anass 
 |-  ( ( ch /\ n = suc m /\ p = suc n ) <-> ( ch /\ ( n = suc m /\ p = suc n ) ) )
28 23 26 27 sylanbrc 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( ch /\ n = suc m /\ p = suc n ) )
29 biid 
 |-  ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
30 biid 
 |-  ( ( n e. D /\ p = suc n /\ m e. n ) <-> ( n e. D /\ p = suc n /\ m e. n ) )
31 1 2 3 13 14 29 30 bnj969 
 |-  ( ( ( R _FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
32 22 28 31 syl2anc 
 |-  ( ( th /\ ch /\ ta /\ et ) -> C e. _V )
33 3 bnj1235 
 |-  ( ch -> f Fn n )
34 33 bnj706 
 |-  ( ( th /\ ch /\ ta /\ et ) -> f Fn n )
35 5 simp3bi 
 |-  ( ta -> p = suc n )
36 35 bnj707 
 |-  ( ( th /\ ch /\ ta /\ et ) -> p = suc n )
37 6 simplbi 
 |-  ( et -> i e. n )
38 37 bnj708 
 |-  ( ( th /\ ch /\ ta /\ et ) -> i e. n )
39 32 34 36 38 bnj951 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) )
40 15 bnj945 
 |-  ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) -> ( G ` i ) = ( f ` i ) )
41 39 40 syl 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( G ` i ) = ( f ` i ) )
42 18 41 eleqtrrd 
 |-  ( ( th /\ ch /\ ta /\ et ) -> y e. ( G ` i ) )
43 16 anim1i 
 |-  ( ( ( th /\ ch /\ ta /\ et ) /\ y e. ( G ` i ) ) -> ( ( ch" /\ i e. _om /\ suc i e. p ) /\ y e. ( G ` i ) ) )
44 df-bnj17 
 |-  ( ( ch" /\ i e. _om /\ suc i e. p /\ y e. ( G ` i ) ) <-> ( ( ch" /\ i e. _om /\ suc i e. p ) /\ y e. ( G ` i ) ) )
45 43 44 sylibr 
 |-  ( ( ( th /\ ch /\ ta /\ et ) /\ y e. ( G ` i ) ) -> ( ch" /\ i e. _om /\ suc i e. p /\ y e. ( G ` i ) ) )
46 1 2 3 7 8 9 10 11 12 14 15 bnj999 
 |-  ( ( ch" /\ i e. _om /\ suc i e. p /\ y e. ( G ` i ) ) -> _pred ( y , A , R ) C_ ( G ` suc i ) )
47 45 46 syl 
 |-  ( ( ( th /\ ch /\ ta /\ et ) /\ y e. ( G ` i ) ) -> _pred ( y , A , R ) C_ ( G ` suc i ) )
48 42 47 mpdan 
 |-  ( ( th /\ ch /\ ta /\ et ) -> _pred ( y , A , R ) C_ ( G ` suc i ) )