Metamath Proof Explorer

Theorem bnj1018g

Description: Version of bnj1018 with less disjoint variable conditions, but requiring ax-13 . (Contributed by Gino Giotto, 27-Mar-2024) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1018.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj1018.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj1018.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1018.4 
|- ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
bnj1018.5 
|- ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
bnj1018.7 
|- ( ph' <-> [. p / n ]. ph )
bnj1018.8 
|- ( ps' <-> [. p / n ]. ps )
bnj1018.9 
|- ( ch' <-> [. p / n ]. ch )
bnj1018.10 
|- ( ph" <-> [. G / f ]. ph' )
bnj1018.11 
|- ( ps" <-> [. G / f ]. ps' )
bnj1018.12 
|- ( ch" <-> [. G / f ]. ch' )
bnj1018.13 
|- D = ( _om \ { (/) } )
bnj1018.14 
|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1018.15 
|- C = U_ y e. ( f ` m ) _pred ( y , A , R )
bnj1018.16 
|- G = ( f u. { <. n , C >. } )
bnj1018.26 
|- ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
bnj1018.29 
|- ( ( th /\ ch /\ ta /\ et ) -> ch" )
bnj1018.30 
|- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. _om /\ suc i e. p ) )
Assertion bnj1018g 
|- ( ( th /\ ch /\ et /\ E. p ta ) -> ( G ` suc i ) C_ _trCl ( X , A , R ) )

Proof

Step Hyp Ref Expression
1 bnj1018.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj1018.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj1018.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj1018.4 
 |-  ( th <-> ( R _FrSe A /\ X e. A /\ y e. _trCl ( X , A , R ) /\ z e. _pred ( y , A , R ) ) )
5 bnj1018.5 
 |-  ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
6 bnj1018.7 
 |-  ( ph' <-> [. p / n ]. ph )
7 bnj1018.8 
 |-  ( ps' <-> [. p / n ]. ps )
8 bnj1018.9 
 |-  ( ch' <-> [. p / n ]. ch )
9 bnj1018.10 
 |-  ( ph" <-> [. G / f ]. ph' )
10 bnj1018.11 
 |-  ( ps" <-> [. G / f ]. ps' )
11 bnj1018.12 
 |-  ( ch" <-> [. G / f ]. ch' )
12 bnj1018.13 
 |-  D = ( _om \ { (/) } )
13 bnj1018.14 
 |-  B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
14 bnj1018.15 
 |-  C = U_ y e. ( f ` m ) _pred ( y , A , R )
15 bnj1018.16 
 |-  G = ( f u. { <. n , C >. } )
16 bnj1018.26 
 |-  ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
17 bnj1018.29 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ch" )
18 bnj1018.30 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. _om /\ suc i e. p ) )
19 df-bnj17 
 |-  ( ( th /\ ch /\ et /\ E. p ta ) <-> ( ( th /\ ch /\ et ) /\ E. p ta ) )
20 bnj258 
 |-  ( ( th /\ ch /\ ta /\ et ) <-> ( ( th /\ ch /\ et ) /\ ta ) )
21 20 17 sylbir 
 |-  ( ( ( th /\ ch /\ et ) /\ ta ) -> ch" )
22 21 ex 
 |-  ( ( th /\ ch /\ et ) -> ( ta -> ch" ) )
23 22 eximdv 
 |-  ( ( th /\ ch /\ et ) -> ( E. p ta -> E. p ch" ) )
24 3 8 11 13 15 bnj985 
 |-  ( G e. B <-> E. p ch" )
25 23 24 syl6ibr 
 |-  ( ( th /\ ch /\ et ) -> ( E. p ta -> G e. B ) )
26 25 imp 
 |-  ( ( ( th /\ ch /\ et ) /\ E. p ta ) -> G e. B )
27 19 26 sylbi 
 |-  ( ( th /\ ch /\ et /\ E. p ta ) -> G e. B )
28 bnj1019 
 |-  ( E. p ( th /\ ch /\ ta /\ et ) <-> ( th /\ ch /\ et /\ E. p ta ) )
29 18 simp3d 
 |-  ( ( th /\ ch /\ ta /\ et ) -> suc i e. p )
30 16 bnj1235 
 |-  ( ch" -> G Fn p )
31 fndm 
 |-  ( G Fn p -> dom G = p )
32 17 30 31 3syl 
 |-  ( ( th /\ ch /\ ta /\ et ) -> dom G = p )
33 29 32 eleqtrrd 
 |-  ( ( th /\ ch /\ ta /\ et ) -> suc i e. dom G )
34 33 exlimiv 
 |-  ( E. p ( th /\ ch /\ ta /\ et ) -> suc i e. dom G )
35 28 34 sylbir 
 |-  ( ( th /\ ch /\ et /\ E. p ta ) -> suc i e. dom G )
36 15 bnj918 
 |-  G e. _V
37 vex 
 |-  i e. _V
38 37 sucex 
 |-  suc i e. _V
39 1 2 12 13 36 38 bnj1015 
 |-  ( ( G e. B /\ suc i e. dom G ) -> ( G ` suc i ) C_ _trCl ( X , A , R ) )
40 27 35 39 syl2anc 
 |-  ( ( th /\ ch /\ et /\ E. p ta ) -> ( G ` suc i ) C_ _trCl ( X , A , R ) )