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Theorem bnj910 28980
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
bnj910.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj910.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj910.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj910.4  |-  ( ph'  <->  [. p  /  n ]. ph )
bnj910.5  |-  ( ps'  <->  [. p  /  n ]. ps )
bnj910.6  |-  ( ch'  <->  [. p  /  n ]. ch )
bnj910.7  |-  ( ph"  <->  [. G  / 
f ]. ph' )
bnj910.8  |-  ( ps"  <->  [. G  / 
f ]. ps' )
bnj910.9  |-  ( ch"  <->  [. G  / 
f ]. ch' )
bnj910.10  |-  D  =  ( om  \  { (/)
} )
bnj910.11  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj910.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj910.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj910.14  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj910.15  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
Assertion
Ref Expression
bnj910  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ch" )
Distinct variable groups:    A, f,
i, m, n, y    D, f, i, n    i, G    R, f, i, m, n, y    f, X, i, n    f, p, i, n    ph, i
Allowed substitution hints:    ph( y, f, m, n, p)    ps( y, f, i, m, n, p)    ch( y, f, i, m, n, p)    ta( y, f, i, m, n, p)    si( y, f, i, m, n, p)    A( p)    B( y, f, i, m, n, p)    C( y, f, i, m, n, p)    D( y, m, p)    R( p)    G( y, f, m, n, p)    X( y, m, p)    ph'( y, f, i, m, n, p)    ps'( y, f, i, m, n, p)    ch'( y, f, i, m, n, p)    ph"( y, f, i, m, n, p)   
ps"( y, f, i, m, n, p)    ch"( y, f, i, m, n, p)

Proof of Theorem bnj910
StepHypRef Expression
1 bnj910.3 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj910.10 . . . 4  |-  D  =  ( om  \  { (/)
} )
31, 2bnj970 28979 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  e.  D )
4 bnj910.1 . . . . 5  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
5 bnj910.2 . . . . 5  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
6 bnj910.12 . . . . 5  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
7 bnj910.14 . . . . 5  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
8 bnj910.15 . . . . 5  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
94, 5, 1, 2, 6, 7, 8bnj969 28978 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
10 simpr3 963 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  =  suc  n )
111bnj1235 28837 . . . . . 6  |-  ( ch 
->  f  Fn  n
)
12113ad2ant1 976 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
1312adantl 452 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
f  Fn  n )
14 bnj910.13 . . . . . 6  |-  G  =  ( f  u.  { <. n ,  C >. } )
1514bnj941 28804 . . . . 5  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )
16153impib 1149 . . . 4  |-  ( ( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n )  ->  G  Fn  p )
179, 10, 13, 16syl3anc 1182 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  G  Fn  p )
18 bnj910.4 . . . 4  |-  ( ph'  <->  [. p  /  n ]. ph )
19 bnj910.7 . . . 4  |-  ( ph"  <->  [. G  / 
f ]. ph' )
204, 5, 1, 18, 19, 2, 6, 14, 7, 8bnj944 28970 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ph" )
21 bnj910.5 . . . 4  |-  ( ps'  <->  [. p  /  n ]. ps )
22 bnj910.8 . . . 4  |-  ( ps"  <->  [. G  / 
f ]. ps' )
235, 1, 2, 6, 14, 9bnj967 28977 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
241, 2, 6, 14, 9, 17bnj966 28976 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
255, 1, 21, 22, 6, 14, 23, 24bnj964 28975 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ps" )
263, 17, 20, 25bnj951 28807 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )
27 bnj910.6 . . . 4  |-  ( ch'  <->  [. p  /  n ]. ch )
28 vex 2791 . . . 4  |-  p  e. 
_V
291, 18, 21, 27, 28bnj919 28797 . . 3  |-  ( ch'  <->  (
p  e.  D  /\  f  Fn  p  /\  ph' 
/\  ps' ) )
30 bnj910.9 . . 3  |-  ( ch"  <->  [. G  / 
f ]. ch' )
3114bnj918 28796 . . 3  |-  G  e. 
_V
3229, 19, 22, 30, 31bnj976 28809 . 2  |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )
3326, 32sylibr 203 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ch" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   [.wsbc 2991    \ cdif 3149    u. cun 3150   (/)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
This theorem is referenced by:  bnj998  28988
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
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