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Theorem bnj944 29742
Description: Technical lemma for bnj69 29812. 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 459 . . . 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 29555 . . . . . . . 8  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
f  Fn  n  /\  ph 
/\  ps ) )
42, 3sylbi 199 . . . . . . 7  |-  ( ch 
->  ( f  Fn  n  /\  ph  /\  ps )
)
5 bnj944.14 . . . . . . 7  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
64, 5sylibr 216 . . . . . 6  |-  ( ch 
->  ta )
763ad2ant1 1028 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ta )
87adantl 468 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ta )
92bnj1232 29608 . . . . . . 7  |-  ( ch 
->  n  e.  D
)
10 vex 3047 . . . . . . . 8  |-  m  e. 
_V
1110bnj216 29533 . . . . . . 7  |-  ( n  =  suc  m  ->  m  e.  n )
12 id 22 . . . . . . 7  |-  ( p  =  suc  n  ->  p  =  suc  n )
139, 11, 123anim123i 1192 . . . . . 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 993 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1614, 15bitri 253 . . . . . 6  |-  ( si  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1713, 16sylibr 216 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  si )
1817adantl 468 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  si )
19 bnj253 29502 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ta  /\ 
si ) )
201, 8, 18, 19syl3anbrc 1191 . . 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 29741 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )  e.  _V )
2621, 25syl5eqel 2532 . . 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 29554 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
n  e.  D  /\  f  Fn  n  /\  ph ) )
292, 28sylbi 199 . . . . 5  |-  ( ch 
->  ( n  e.  D  /\  f  Fn  n  /\  ph ) )
30293ad2ant1 1028 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  f  Fn  n  /\  ph ) )
31 simp3 1009 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  p  =  suc  n )
32 bnj291 29509 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n  /\  ph )  /\  p  =  suc  n ) )
3330, 31, 32sylanbrc 669 . . 3  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph ) )
3433adantl 468 . 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 4166 . . . . . . . . 9  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  <. n ,  C >.  =  <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. )
3837sneqd 3979 . . . . . . . 8  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  { <. n ,  C >. }  =  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
3938uneq2d 3587 . . . . . . 7  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
f  u.  { <. n ,  C >. } )  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
4036, 39syl5eq 2496 . . . . . 6  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
4140sbceq1d 3271 . . . . 5  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( [. G  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4235, 41syl5bb 261 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( ph"  <->  [. ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' ) )
4342imbi2d 318 . . 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 240 . . . 4  |-  ( [. ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph'  <->  [. ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  /  f ]. ph' )
46 eqid 2450 . . . 4  |-  ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  =  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
47 0ex 4534 . . . . 5  |-  (/)  e.  _V
4847elimel 3942 . . . 4  |-  if ( C  e.  _V ,  C ,  (/) )  e. 
_V
4923, 44, 45, 22, 46, 48bnj929 29740 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  [. ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  / 
f ]. ph' )
5043, 49dedth 3931 . 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   _Vcvv 3044   [.wsbc 3266    \ cdif 3400    u. cun 3401   (/)c0 3730   ifcif 3880   {csn 3967   <.cop 3973   U_ciun 4277   suc csuc 5424    Fn wfn 5576   ` cfv 5581   omcom 6689    /\ w-bnj17 29484    predc-bnj14 29486    FrSe w-bnj15 29490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580  ax-reg 8104
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-bnj17 29485  df-bnj14 29487  df-bnj13 29489  df-bnj15 29491
This theorem is referenced by:  bnj910  29752
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