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Theorem bnj1006 29307
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
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
Ref Expression
bnj1006  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  pred ( y ,  A ,  R )  C_  ( G `  suc  i ) )
Distinct variable groups:    A, f,
i, m, n, y    D, f, n    i, G    R, f, i, m, n, y    f, X, n   
f, p, i, n
Allowed substitution hints:    ph( y, z, f, i, m, n, p)    ps( y, z, f, i, m, n, p)    ch( y, z, f, i, m, n, p)    th( y,
z, f, i, m, n, p)    ta( y,
z, f, i, m, n, p)    et( y,
z, f, i, m, n, p)    A( z, p)    C( y, z, f, i, m, n, p)    D( y, z, i, m, p)    R( z, p)    G( y, z, f, m, n, p)    X( y, z, i, 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 bnj1006
StepHypRef Expression
1 bnj1006.6 . . . . 5  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
21simprbi 450 . . . 4  |-  ( et 
->  y  e.  (
f `  i )
)
32bnj708 29101 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
y  e.  ( f `
 i ) )
4 bnj1006.4 . . . . . . . 8  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) ) )
5 bnj253 29045 . . . . . . . . 9  |-  ( ( 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 ) ) )
65simp1bi 970 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) )  -> 
( R  FrSe  A  /\  X  e.  A
) )
74, 6sylbi 187 . . . . . . 7  |-  ( th 
->  ( R  FrSe  A  /\  X  e.  A
) )
87bnj705 29098 . . . . . 6  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( R  FrSe  A  /\  X  e.  A
) )
9 bnj643 29094 . . . . . . 7  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch )
10 bnj1006.5 . . . . . . . . 9  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
11 3simpc 954 . . . . . . . . 9  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ( n  =  suc  m  /\  p  =  suc  n ) )
1210, 11sylbi 187 . . . . . . . 8  |-  ( ta 
->  ( n  =  suc  m  /\  p  =  suc  n ) )
1312bnj707 29100 . . . . . . 7  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( n  =  suc  m  /\  p  =  suc  n ) )
14 3anass 938 . . . . . . 7  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  <->  ( ch  /\  ( n  =  suc  m  /\  p  =  suc  n ) ) )
159, 13, 14sylanbrc 645 . . . . . 6  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )
16 bnj1006.1 . . . . . . 7  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
17 bnj1006.2 . . . . . . 7  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
18 bnj1006.3 . . . . . . 7  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
19 bnj1006.13 . . . . . . 7  |-  D  =  ( om  \  { (/)
} )
20 bnj1006.15 . . . . . . 7  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
21 biid 227 . . . . . . 7  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
22 biid 227 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
2316, 17, 18, 19, 20, 21, 22bnj969 29294 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
248, 15, 23syl2anc 642 . . . . 5  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  C  e.  _V )
2518bnj1235 29153 . . . . . 6  |-  ( ch 
->  f  Fn  n
)
2625bnj706 29099 . . . . 5  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
f  Fn  n )
2710simp3bi 972 . . . . . 6  |-  ( ta 
->  p  =  suc  n )
2827bnj707 29100 . . . . 5  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  p  =  suc  n )
291simplbi 446 . . . . . 6  |-  ( et 
->  i  e.  n
)
3029bnj708 29101 . . . . 5  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  n )
3124, 26, 28, 30bnj951 29123 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )
)
32 bnj1006.16 . . . . 5  |-  G  =  ( f  u.  { <. n ,  C >. } )
3332bnj945 29121 . . . 4  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )  ->  ( G `  i
)  =  ( f `
 i ) )
3431, 33syl 15 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( G `  i
)  =  ( f `
 i ) )
353, 34eleqtrrd 2373 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
y  e.  ( G `
 i ) )
36 bnj1006.28 . . . . 5  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch"  /\  i  e. 
om  /\  suc  i  e.  p ) )
3736anim1i 551 . . . 4  |-  ( ( ( th  /\  ch  /\ 
ta  /\  et )  /\  y  e.  ( G `  i )
)  ->  ( ( ch" 
/\  i  e.  om  /\ 
suc  i  e.  p
)  /\  y  e.  ( G `  i ) ) )
38 df-bnj17 29028 . . . 4  |-  ( ( ch" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  <->  ( ( ch"  /\  i  e.  om  /\  suc  i  e.  p )  /\  y  e.  ( G `  i
) ) )
3937, 38sylibr 203 . . 3  |-  ( ( ( th  /\  ch  /\ 
ta  /\  et )  /\  y  e.  ( G `  i )
)  ->  ( ch"  /\  i  e.  om  /\  suc  i  e.  p  /\  y  e.  ( G `  i
) ) )
40 bnj1006.7 . . . 4  |-  ( ph'  <->  [. p  /  n ]. ph )
41 bnj1006.8 . . . 4  |-  ( ps'  <->  [. p  /  n ]. ps )
42 bnj1006.9 . . . 4  |-  ( ch'  <->  [. p  /  n ]. ch )
43 bnj1006.10 . . . 4  |-  ( ph"  <->  [. G  / 
f ]. ph' )
44 bnj1006.11 . . . 4  |-  ( ps"  <->  [. G  / 
f ]. ps' )
45 bnj1006.12 . . . 4  |-  ( ch"  <->  [. G  / 
f ]. ch' )
4616, 17, 18, 40, 41, 42, 43, 44, 45, 20, 32bnj999 29305 . . 3  |-  ( ( ch" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  ->  pred ( y ,  A ,  R
)  C_  ( G `  suc  i ) )
4739, 46syl 15 . 2  |-  ( ( ( th  /\  ch  /\ 
ta  /\  et )  /\  y  e.  ( G `  i )
)  ->  pred ( y ,  A ,  R
)  C_  ( G `  suc  i ) )
4835, 47mpdan 649 1  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  pred ( y ,  A ,  R )  C_  ( G `  suc  i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _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    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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-reg 7322
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034
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