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Theorem bnj944 28970
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
bnj944.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj944.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj944.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj944.4  |-  ( ph'  <->  [. p  /  n ]. ph )
bnj944.7  |-  ( ph"  <->  [. G  / 
f ]. ph' )
bnj944.10  |-  D  =  ( om  \  { (/)
} )
bnj944.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj944.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj944.14  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj944.15  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
Assertion
Ref Expression
bnj944  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ph" )
Distinct variable groups:    A, f,
i, m, n    y, A, f, i, m    R, f, i, m, n    y, R    f, X, n
Allowed substitution hints:    ph( y, f, i, 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)    C( y,
f, i, m, n, p)    D( y, f, i, m, n, p)    R( p)    G( y, f, i, m, n, p)    X( y, i, m, p)    ph'( y, f, i, m, n, p)    ph"( y, f, i, m, n, p)

Proof of Theorem bnj944
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( R  FrSe  A  /\  X  e.  A
) )
2 bnj944.3 . . . . . . . 8  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
3 bnj667 28781 . . . . . . . 8  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
f  Fn  n  /\  ph 
/\  ps ) )
42, 3sylbi 187 . . . . . . 7  |-  ( ch 
->  ( f  Fn  n  /\  ph  /\  ps )
)
5 bnj944.14 . . . . . . 7  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
64, 5sylibr 203 . . . . . 6  |-  ( ch 
->  ta )
763ad2ant1 976 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ta )
87adantl 452 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ta )
92bnj1232 28836 . . . . . . 7  |-  ( ch 
->  n  e.  D
)
10 vex 2791 . . . . . . . 8  |-  m  e. 
_V
1110bnj216 28760 . . . . . . 7  |-  ( n  =  suc  m  ->  m  e.  n )
12 id 19 . . . . . . 7  |-  ( p  =  suc  n  ->  p  =  suc  n )
139, 11, 123anim123i 1137 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
14 bnj944.15 . . . . . . 7  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
15 3ancomb 943 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1614, 15bitri 240 . . . . . 6  |-  ( si  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1713, 16sylibr 203 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  si )
1817adantl 452 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  si )
19 bnj253 28729 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ta  /\ 
si ) )
201, 8, 18, 19syl3anbrc 1136 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )
)
21 bnj944.12 . . . 4  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
22 bnj944.10 . . . . 5  |-  D  =  ( om  \  { (/)
} )
23 bnj944.1 . . . . 5  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
24 bnj944.2 . . . . 5  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2522, 5, 14, 23, 24bnj938 28969 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )  e.  _V )
2621, 25syl5eqel 2367 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  C  e.  _V )
2720, 26syl 15 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
28 bnj658 28780 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
n  e.  D  /\  f  Fn  n  /\  ph ) )
292, 28sylbi 187 . . . . 5  |-  ( ch 
->  ( n  e.  D  /\  f  Fn  n  /\  ph ) )
30293ad2ant1 976 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  f  Fn  n  /\  ph ) )
31 simp3 957 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  p  =  suc  n )
32 bnj291 28736 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n  /\  ph )  /\  p  =  suc  n ) )
3330, 31, 32sylanbrc 645 . . 3  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph ) )
3433adantl 452 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph ) )
35 bnj944.7 . . . . 5  |-  ( ph"  <->  [. G  / 
f ]. ph' )
36 bnj944.13 . . . . . . 7  |-  G  =  ( f  u.  { <. n ,  C >. } )
37 opeq2 3797 . . . . . . . . 9  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  <. n ,  C >.  =  <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. )
3837sneqd 3653 . . . . . . . 8  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  { <. n ,  C >. }  =  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
3938uneq2d 3329 . . . . . . 7  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
f  u.  { <. n ,  C >. } )  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
4036, 39syl5eq 2327 . . . . . 6  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
41 dfsbcq 2993 . . . . . 6  |-  ( G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  ->  ( [. G  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4240, 41syl 15 . . . . 5  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( [. G  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4335, 42syl5bb 248 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( ph"  <->  [. ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4443imbi2d 307 . . 3  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" )  <->  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) ) )
45 bnj944.4 . . . 4  |-  ( ph'  <->  [. p  /  n ]. ph )
46 biid 227 . . . 4  |-  ( [. ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' )
47 eqid 2283 . . . 4  |-  ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  =  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
48 0ex 4150 . . . . 5  |-  (/)  e.  _V
4948elimel 3617 . . . 4  |-  if ( C  e.  _V ,  C ,  (/) )  e. 
_V
5023, 45, 46, 22, 47, 49bnj929 28968 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  [. ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  / 
f ]. ph' )
5144, 50dedth 3606 . 2  |-  ( C  e.  _V  ->  (
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" ) )
5227, 34, 51sylc 56 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ph" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991    \ cdif 3149    u. cun 3150   (/)c0 3455   ifcif 3565   {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:  bnj910  28980
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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