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Theorem bnj557 28978
Description: Technical lemma for bnj852 28998. 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 df-bnj17 28757 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze ) 
<->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze ) )
2 bnj256 28776 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze ) 
<->  ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze )
) )
31, 2bitr3i 243 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze )  <->  ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze ) ) )
4 bnj557.18 . . . . . . . 8  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
5 bnj557.19 . . . . . . . 8  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
64, 5bnj556 28977 . . . . . . 7  |-  ( et 
->  si )
763anim3i 1141 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( R  FrSe  A  /\  ta  /\  si )
)
8 bnj557.20 . . . . . . 7  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
9 vex 2919 . . . . . . . 8  |-  i  e. 
_V
109bnj216 28805 . . . . . . 7  |-  ( m  =  suc  i  -> 
i  e.  m )
118, 10bnj837 28836 . . . . . 6  |-  ( ze 
->  i  e.  m
)
127, 11anim12i 550 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze )  -> 
( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
) )
133, 12sylbir 205 . . . 4  |-  ( ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze ) )  ->  ( ( R 
FrSe  A  /\  ta  /\  si )  /\  i  e.  m ) )
145bnj1254 28887 . . . . . 6  |-  ( et 
->  m  =  suc  p )
158simp3bi 974 . . . . . 6  |-  ( ze 
->  m  =  suc  i )
16 bnj551 28816 . . . . . 6  |-  ( ( m  =  suc  p  /\  m  =  suc  i )  ->  p  =  i )
1714, 15, 16syl2an 464 . . . . 5  |-  ( ( et  /\  ze )  ->  p  =  i )
1817adantl 453 . . . 4  |-  ( ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze ) )  ->  p  =  i )
1913, 18jca 519 . . 3  |-  ( ( ( R  FrSe  A  /\  ta )  /\  ( et  /\  ze ) )  ->  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m )  /\  p  =  i
) )
20 df-3an 938 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  <->  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m )  /\  p  =  i
) )
2119, 2, 203imtr4i 258 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( ( R 
FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i ) )
22 bnj557.28 . . 3  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
23 bnj557.29 . . 3  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
24 bnj557.3 . . 3  |-  D  =  ( om  \  { (/)
} )
25 bnj557.16 . . 3  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
26 bnj557.17 . . 3  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
27 bnj557.22 . . 3  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
28 bnj557.25 . . 3  |-  G  =  ( f  u.  { <. m ,  C >. } )
29 bnj557.21 . . 3  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
30 bnj557.23 . . 3  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
31 bnj557.24 . . 3  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
32 bnj557.36 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
3322, 23, 24, 25, 26, 4, 27, 28, 29, 30, 31, 32bnj553 28975 . 2  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  ( G `  m )  =  L )
3421, 33syl 16 1  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( G `  m )  =  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    \ cdif 3277    u. cun 3278   (/)c0 3588   {csn 3774   <.cop 3777   U_ciun 4053   suc csuc 4543   omcom 4804    Fn wfn 5408   ` cfv 5413    /\ w-bnj17 28756    predc-bnj14 28758    FrSe w-bnj15 28762
This theorem is referenced by:  bnj558  28979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660  ax-reg 7516
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-eprel 4454  df-id 4458  df-fr 4501  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-bnj17 28757
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