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Theorem bnj944 29578
Description: Technical lemma for bnj69 29648. 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 458 . . . 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 29391 . . . . . . . 8  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
f  Fn  n  /\  ph 
/\  ps ) )
42, 3sylbi 198 . . . . . . 7  |-  ( ch 
->  ( f  Fn  n  /\  ph  /\  ps )
)
5 bnj944.14 . . . . . . 7  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
64, 5sylibr 215 . . . . . 6  |-  ( ch 
->  ta )
763ad2ant1 1026 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ta )
87adantl 467 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ta )
92bnj1232 29444 . . . . . . 7  |-  ( ch 
->  n  e.  D
)
10 vex 3081 . . . . . . . 8  |-  m  e. 
_V
1110bnj216 29369 . . . . . . 7  |-  ( n  =  suc  m  ->  m  e.  n )
12 id 23 . . . . . . 7  |-  ( p  =  suc  n  ->  p  =  suc  n )
139, 11, 123anim123i 1190 . . . . . 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 991 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1614, 15bitri 252 . . . . . 6  |-  ( si  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1713, 16sylibr 215 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  si )
1817adantl 467 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  si )
19 bnj253 29338 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ta  /\ 
si ) )
201, 8, 18, 19syl3anbrc 1189 . . 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 29577 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )  e.  _V )
2621, 25syl5eqel 2512 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  C  e.  _V )
2720, 26syl 17 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
28 bnj658 29390 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
n  e.  D  /\  f  Fn  n  /\  ph ) )
292, 28sylbi 198 . . . . 5  |-  ( ch 
->  ( n  e.  D  /\  f  Fn  n  /\  ph ) )
30293ad2ant1 1026 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  f  Fn  n  /\  ph ) )
31 simp3 1007 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  p  =  suc  n )
32 bnj291 29345 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n  /\  ph )  /\  p  =  suc  n ) )
3330, 31, 32sylanbrc 668 . . 3  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph ) )
3433adantl 467 . 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 4182 . . . . . . . . 9  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  <. n ,  C >.  =  <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. )
3837sneqd 4005 . . . . . . . 8  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  { <. n ,  C >. }  =  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
3938uneq2d 3617 . . . . . . 7  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
f  u.  { <. n ,  C >. } )  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
4036, 39syl5eq 2473 . . . . . 6  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
4140sbceq1d 3301 . . . . 5  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( [. G  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4235, 41syl5bb 260 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( ph"  <->  [. ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4342imbi2d 317 . . 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' ) ) )
44 bnj944.4 . . . 4  |-  ( ph'  <->  [. p  /  n ]. ph )
45 biid 239 . . . 4  |-  ( [. ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' )
46 eqid 2420 . . . 4  |-  ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  =  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
47 0ex 4548 . . . . 5  |-  (/)  e.  _V
4847elimel 3968 . . . 4  |-  if ( C  e.  _V ,  C ,  (/) )  e. 
_V
4923, 44, 45, 22, 46, 48bnj929 29576 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  [. ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  / 
f ]. ph' )
5043, 49dedth 3957 . 2  |-  ( C  e.  _V  ->  (
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" ) )
5127, 34, 50sylc 62 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ph" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078   [.wsbc 3296    \ cdif 3430    u. cun 3431   (/)c0 3758   ifcif 3906   {csn 3993   <.cop 3999   U_ciun 4293   suc csuc 5435    Fn wfn 5587   ` cfv 5592   omcom 6697    /\ w-bnj17 29320    predc-bnj14 29322    FrSe w-bnj15 29326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588  ax-reg 8098
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-bnj17 29321  df-bnj14 29323  df-bnj13 29325  df-bnj15 29327
This theorem is referenced by:  bnj910  29588
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