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Theorem bnj557 34081
Description: Technical lemma for bnj852 34101. 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
bnj557.3  |-  D  =  ( om  \  { (/)
} )
bnj557.16  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj557.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj557.18  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj557.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj557.20  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
bnj557.21  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
bnj557.22  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
bnj557.23  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj557.24  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
bnj557.25  |-  G  =  ( f  u.  { <. m ,  C >. } )
bnj557.28  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj557.29  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj557.36  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Assertion
Ref Expression
bnj557  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( G `  m )  =  L )
Distinct variable groups:    A, i, p, y    y, G    R, i, p, y    f, i, p, y    i, m, p    p, ph'
Allowed substitution hints:    ta( x, y, f, i, m, n, p)    et( x, y, f, i, m, n, p)    ze( x, y, f, i, m, n, p)    si( x, y, f, i, m, n, p)    A( x, f, m, n)    B( x, y, f, i, m, n, p)    C( x, y, f, i, m, n, p)    D( x, y, f, i, m, n, p)    R( x, f, m, n)    G( x, f, i, m, n, p)    K( x, y, f, i, m, n, p)    L( x, y, f, i, m, n, p)    ph'( x, y, f, i, m, n)    ps'( x, y, f, i, m, n, p)

Proof of Theorem bnj557
StepHypRef Expression
1 3an4anass 1219 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze )  <->  ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze ) ) )
2 bnj557.18 . . . . . . . 8  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
3 bnj557.19 . . . . . . . 8  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
42, 3bnj556 34080 . . . . . . 7  |-  ( et 
->  si )
543anim3i 1184 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( R  FrSe  A  /\  ta  /\  si )
)
6 bnj557.20 . . . . . . 7  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
7 vex 3112 . . . . . . . 8  |-  i  e. 
_V
87bnj216 33909 . . . . . . 7  |-  ( m  =  suc  i  -> 
i  e.  m )
96, 8bnj837 33941 . . . . . 6  |-  ( ze 
->  i  e.  m
)
105, 9anim12i 566 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze )  -> 
( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
) )
111, 10sylbir 213 . . . 4  |-  ( ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze ) )  ->  ( ( R 
FrSe  A  /\  ta  /\  si )  /\  i  e.  m ) )
123bnj1254 33990 . . . . . 6  |-  ( et 
->  m  =  suc  p )
136simp3bi 1013 . . . . . 6  |-  ( ze 
->  m  =  suc  i )
14 bnj551 33921 . . . . . 6  |-  ( ( m  =  suc  p  /\  m  =  suc  i )  ->  p  =  i )
1512, 13, 14syl2an 477 . . . . 5  |-  ( ( et  /\  ze )  ->  p  =  i )
1615adantl 466 . . . 4  |-  ( ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze ) )  ->  p  =  i )
1711, 16jca 532 . . 3  |-  ( ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze ) )  ->  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m )  /\  p  =  i
) )
18 bnj256 33880 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze ) 
<->  ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze )
) )
19 df-3an 975 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  <->  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m )  /\  p  =  i
) )
2017, 18, 193imtr4i 266 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( ( R 
FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i ) )
21 bnj557.28 . . 3  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
22 bnj557.29 . . 3  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
23 bnj557.3 . . 3  |-  D  =  ( om  \  { (/)
} )
24 bnj557.16 . . 3  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
25 bnj557.17 . . 3  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
26 bnj557.22 . . 3  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
27 bnj557.25 . . 3  |-  G  =  ( f  u.  { <. m ,  C >. } )
28 bnj557.21 . . 3  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
29 bnj557.23 . . 3  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
30 bnj557.24 . . 3  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
31 bnj557.36 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
3221, 22, 23, 24, 25, 2, 26, 27, 28, 29, 30, 31bnj553 34078 . 2  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  ( G `  m )  =  L )
3320, 32syl 16 1  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( G `  m )  =  L )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    \ cdif 3468    u. cun 3469   (/)c0 3793   {csn 4032   <.cop 4038   U_ciun 4332   suc csuc 4889    Fn wfn 5589   ` cfv 5594   omcom 6699    /\ w-bnj17 33860    predc-bnj14 33862    FrSe w-bnj15 33866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591  ax-reg 8036
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-eprel 4800  df-id 4804  df-fr 4847  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-bnj17 33861
This theorem is referenced by:  bnj558  34082
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