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Theorem bnj907 28997
Description: Technical lemma for bnj69 29040. 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
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
Ref Expression
bnj907  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
Distinct variable groups:    A, f,
i, m, n, p, y    z, A, y    D, f, i, n    i, G, p    R, f, i, m, n, p, y   
z, R    f, X, i, m, n, y    z, X    ch, m, p    et, m, p    th, f, i, m, n, p    ph, i
Allowed substitution hints:    ph( y, z, f, m, n, p)    ps( y, z, f, i, m, n, p)    ch( y, z, f, i, n)    th( y, z)    ta( y,
z, f, i, m, n, p)    et( y,
z, f, i, n)    B( y, z, f, i, m, n, p)    C( y, z, f, i, m, n, p)    D( y,
z, m, p)    G( y, z, f, m, n)    X( 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 bnj907
StepHypRef Expression
1 bnj907.4 . 2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) ) )
2 bnj907.1 . . . . . . . . 9  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
3 bnj907.2 . . . . . . . . 9  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
4 bnj907.3 . . . . . . . . 9  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
5 bnj907.5 . . . . . . . . 9  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
6 bnj907.6 . . . . . . . . 9  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
7 bnj907.13 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
8 bnj907.14 . . . . . . . . 9  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
92, 3, 4, 1, 5, 6, 7, 8bnj1021 28996 . . . . . . . 8  |-  E. f E. n E. i E. m ( th  ->  ( th  /\  ch  /\  et  /\  E. p ta ) )
10 bnj907.7 . . . . . . . . . . . 12  |-  ( ph'  <->  [. p  /  n ]. ph )
11 bnj907.8 . . . . . . . . . . . 12  |-  ( ps'  <->  [. p  /  n ]. ps )
12 bnj907.9 . . . . . . . . . . . 12  |-  ( ch'  <->  [. p  /  n ]. ch )
13 bnj907.10 . . . . . . . . . . . 12  |-  ( ph"  <->  [. G  / 
f ]. ph' )
14 bnj907.11 . . . . . . . . . . . 12  |-  ( ps"  <->  [. G  / 
f ]. ps' )
15 bnj907.12 . . . . . . . . . . . 12  |-  ( ch"  <->  [. G  / 
f ]. ch' )
16 bnj907.15 . . . . . . . . . . . 12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
17 bnj907.16 . . . . . . . . . . . 12  |-  G  =  ( f  u.  { <. n ,  C >. } )
18 vex 2791 . . . . . . . . . . . . . 14  |-  p  e. 
_V
194, 10, 11, 12, 18bnj919 28797 . . . . . . . . . . . . 13  |-  ( ch'  <->  (
p  e.  D  /\  f  Fn  p  /\  ph' 
/\  ps' ) )
2017bnj918 28796 . . . . . . . . . . . . 13  |-  G  e. 
_V
2119, 13, 14, 15, 20bnj976 28809 . . . . . . . . . . . 12  |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )
222, 3, 4, 1, 5, 6, 10, 11, 12, 13, 14, 15, 7, 8, 16, 17, 21bnj1020 28995 . . . . . . . . . . 11  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  ->  pred ( y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) )
2322ax-gen 1533 . . . . . . . . . 10  |-  A. m
( ( th  /\  ch  /\  et  /\  E. p ta )  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
24 19.29r 1584 . . . . . . . . . . 11  |-  ( ( 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 pm3.33 568 . . . . . . . . . . 11  |-  ( ( ( 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 )
) )
2624, 25bnj593 28774 . . . . . . . . . 10  |-  ( ( 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 )
) )
2723, 26mpan2 652 . . . . . . . . 9  |-  ( E. m ( th  ->  ( th  /\  ch  /\  et  /\  E. p ta ) )  ->  E. m
( th  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
28272eximi 1564 . . . . . . . 8  |-  ( 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
) ) )
299, 28bnj101 28749 . . . . . . 7  |-  E. f E. n E. i E. m ( th  ->  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
30 19.9v 1663 . . . . . . 7  |-  ( 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 ) ) )
3129, 30mpbi 199 . . . . . 6  |-  E. n E. i E. m ( th  ->  pred ( y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) )
32 19.9v 1663 . . . . . 6  |-  ( 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 ) ) )
3331, 32mpbi 199 . . . . 5  |-  E. i E. m ( th  ->  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
34 19.9v 1663 . . . . 5  |-  ( 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 ) ) )
3533, 34mpbi 199 . . . 4  |-  E. m
( th  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
36 19.9v 1663 . . . 4  |-  ( E. m ( th  ->  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )  <->  ( th  ->  pred ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
) )
3735, 36mpbi 199 . . 3  |-  ( th 
->  pred ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
)
381bnj1254 28842 . . 3  |-  ( th 
->  z  e.  pred ( y ,  A ,  R ) )
3937, 38sseldd 3181 . 2  |-  ( th 
->  z  e.  trCl ( X ,  A ,  R ) )
401, 39bnj978 28981 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   [.wsbc 2991    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U_ciun 3905   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719    TrFow-bnj19 28721
This theorem is referenced by:  bnj1029  28998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720  df-bnj19 28722
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