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Theorem bnj907 29778
Description: Technical lemma for bnj69 29821. 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 29777 . . . . . . . 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 3085 . . . . . . . . . . . . . 14  |-  p  e. 
_V
194, 10, 11, 12, 18bnj919 29580 . . . . . . . . . . . . 13  |-  ( ch'  <->  (
p  e.  D  /\  f  Fn  p  /\  ph' 
/\  ps' ) )
2017bnj918 29579 . . . . . . . . . . . . 13  |-  G  e. 
_V
2119, 13, 14, 15, 20bnj976 29591 . . . . . . . . . . . 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 29776 . . . . . . . . . . 11  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  ->  pred ( y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) )
2322ax-gen 1666 . . . . . . . . . 10  |-  A. m
( ( th  /\  ch  /\  et  /\  E. p ta )  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
24 19.29r 1730 . . . . . . . . . . 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 pm3.33 588 . . . . . . . . . . 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 29557 . . . . . . . . . 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 676 . . . . . . . . 9  |-  ( E. m ( th  ->  ( th  /\  ch  /\  et  /\  E. p ta ) )  ->  E. m
( th  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
28272eximi 1704 . . . . . . . 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 29531 . . . . . . 7  |-  E. f E. n E. i E. m ( th  ->  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
30 19.9v 1802 . . . . . . 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 212 . . . . . 6  |-  E. n E. i E. m ( th  ->  pred ( y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) )
32 19.9v 1802 . . . . . 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 212 . . . . 5  |-  E. i E. m ( th  ->  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
34 19.9v 1802 . . . . 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 212 . . . 4  |-  E. m
( th  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
36 19.9v 1802 . . . 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 212 . . 3  |-  ( th 
->  pred ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
)
381bnj1254 29623 . . 3  |-  ( th 
->  z  e.  pred ( y ,  A ,  R ) )
3937, 38sseldd 3466 . 2  |-  ( th 
->  z  e.  trCl ( X ,  A ,  R ) )
401, 39bnj978 29762 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983   A.wal 1436    = wceq 1438   E.wex 1660    e. wcel 1869   {cab 2408   A.wral 2776   E.wrex 2777   [.wsbc 3300    \ cdif 3434    u. cun 3435    C_ wss 3437   (/)c0 3762   {csn 3997   <.cop 4003   U_ciun 4297   suc csuc 5442    Fn wfn 5594   ` cfv 5599   omcom 6704    /\ w-bnj17 29493    predc-bnj14 29495    FrSe w-bnj15 29499    trClc-bnj18 29501    TrFow-bnj19 29503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595  ax-reg 8111
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-om 6705  df-bnj17 29494  df-bnj14 29496  df-bnj13 29498  df-bnj15 29500  df-bnj18 29502  df-bnj19 29504
This theorem is referenced by:  bnj1029  29779
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