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Theorem bnj1018 29310
Description: Technical lemma for bnj69 29356. 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.)
Hypotheses
Ref Expression
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
Ref Expression
bnj1018  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  ->  ( G `  suc  i )  C_  trCl ( X ,  A ,  R ) )
Distinct variable groups:    A, f,
i, m, n, y    D, f, i, n    i, G, p    R, f, i, m, n, y    f, X, i, n, y    ch, p    et, p    f, p, n    ph, i    th, p
Allowed substitution hints:    ph( y, z, f, m, n, p)    ps( y, z, f, i, m, n, p)    ch( y, z, f, i, m, n)    th( y, z, f, i, m, n)    ta( y, z, f, i, m, n, p)    et( y,
z, f, i, m, n)    A( z, p)    B( y, z, f, i, m, n, p)    C( y,
z, f, i, m, n, p)    D( y,
z, m, p)    R( z, p)    G( y, z, f, m, n)    X( z, m, p)    ph'( y, z, f, i, m, n, p)    ps'( y, z, f, i, m, n, p)    ch'( y, z, f, i, m, n, p)    ph"( y, z, f, i, m, n, p)   
ps"( y, z, f, i, m, n, p)    ch"( y, z, f, i, m, n, p)

Proof of Theorem bnj1018
StepHypRef Expression
1 df-bnj17 29028 . . 3  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  <->  ( ( th 
/\  ch  /\  et )  /\  E. p ta ) )
2 bnj258 29049 . . . . . . . 8  |-  ( ( th  /\  ch  /\  ta  /\  et )  <->  ( ( th  /\  ch  /\  et )  /\  ta ) )
3 bnj1018.29 . . . . . . . 8  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch" )
42, 3sylbir 204 . . . . . . 7  |-  ( ( ( th  /\  ch  /\  et )  /\  ta )  ->  ch" )
54ex 423 . . . . . 6  |-  ( ( th  /\  ch  /\  et )  ->  ( ta 
->  ch" ) )
65eximdv 1612 . . . . 5  |-  ( ( th  /\  ch  /\  et )  ->  ( E. p ta  ->  E. p ch" ) )
7 bnj1018.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
8 bnj1018.9 . . . . . 6  |-  ( ch'  <->  [. p  /  n ]. ch )
9 bnj1018.12 . . . . . 6  |-  ( ch"  <->  [. G  / 
f ]. ch' )
10 bnj1018.14 . . . . . 6  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
11 bnj1018.16 . . . . . 6  |-  G  =  ( f  u.  { <. n ,  C >. } )
127, 8, 9, 10, 11bnj985 29301 . . . . 5  |-  ( G  e.  B  <->  E. p ch" )
136, 12syl6ibr 218 . . . 4  |-  ( ( th  /\  ch  /\  et )  ->  ( E. p ta  ->  G  e.  B ) )
1413imp 418 . . 3  |-  ( ( ( th  /\  ch  /\  et )  /\  E. p ta )  ->  G  e.  B )
151, 14sylbi 187 . 2  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  ->  G  e.  B )
16 bnj1019 29127 . . 3  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  ( th  /\  ch  /\  et  /\  E. p ta ) )
17 bnj1018.30 . . . . . 6  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch"  /\  i  e. 
om  /\  suc  i  e.  p ) )
1817simp3d 969 . . . . 5  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  suc  i  e.  p
)
19 bnj1018.26 . . . . . . 7  |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )
2019bnj1235 29153 . . . . . 6  |-  ( ch"  ->  G  Fn  p )
21 fndm 5359 . . . . . 6  |-  ( G  Fn  p  ->  dom  G  =  p )
223, 20, 213syl 18 . . . . 5  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  dom  G  =  p )
2318, 22eleqtrrd 2373 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  suc  i  e.  dom  G )
2423exlimiv 1624 . . 3  |-  ( E. p ( th  /\  ch  /\  ta  /\  et )  ->  suc  i  e.  dom  G )
2516, 24sylbir 204 . 2  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  ->  suc  i  e. 
dom  G )
26 bnj1018.1 . . 3  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
27 bnj1018.2 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
28 bnj1018.13 . . 3  |-  D  =  ( om  \  { (/)
} )
2911bnj918 29112 . . 3  |-  G  e. 
_V
30 vex 2804 . . . 4  |-  i  e. 
_V
3130sucex 4618 . . 3  |-  suc  i  e.  _V
3226, 27, 28, 10, 29, 31bnj1015 29309 . 2  |-  ( ( G  e.  B  /\  suc  i  e.  dom  G )  ->  ( G `  suc  i )  C_  trCl ( X ,  A ,  R ) )
3315, 25, 32syl2anc 642 1  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  ->  ( G `  suc  i )  C_  trCl ( X ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801   [.wsbc 3004    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   U_ciun 3921   suc csuc 4410   omcom 4672   dom cdm 4705    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1020  29311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-suc 4414  df-dm 4715  df-iota 5235  df-fn 5274  df-fv 5279  df-bnj17 29028  df-bnj18 29036
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