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Theorem bnj558 29273
Description: Technical lemma for bnj852 29292. 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
bnj558.3  |-  D  =  ( om  \  { (/)
} )
bnj558.16  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj558.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj558.18  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj558.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj558.20  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
bnj558.21  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
bnj558.22  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
bnj558.23  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj558.24  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
bnj558.25  |-  G  =  ( f  u.  { <. m ,  C >. } )
bnj558.28  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj558.29  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj558.36  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Assertion
Ref Expression
bnj558  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( G `  suc  i )  =  K )
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 bnj558
StepHypRef Expression
1 bnj558.3 . . 3  |-  D  =  ( om  \  { (/)
} )
2 bnj558.16 . . 3  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
3 bnj558.17 . . 3  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
4 bnj558.18 . . 3  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
5 bnj558.19 . . 3  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
6 bnj558.20 . . 3  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
7 bnj558.21 . . 3  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
8 bnj558.22 . . 3  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
9 bnj558.23 . . 3  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
10 bnj558.24 . . 3  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
11 bnj558.25 . . 3  |-  G  =  ( f  u.  { <. m ,  C >. } )
12 bnj558.28 . . 3  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
13 bnj558.29 . . 3  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
14 bnj558.36 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14bnj557 29272 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( G `  m )  =  L )
16 bnj422 29079 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze ) 
<->  ( et  /\  ze  /\  R  FrSe  A  /\  ta ) )
17 bnj253 29068 . . . . 5  |-  ( ( et  /\  ze  /\  R  FrSe  A  /\  ta ) 
<->  ( ( et  /\  ze )  /\  R  FrSe  A  /\  ta ) )
1816, 17bitri 241 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze ) 
<->  ( ( et  /\  ze )  /\  R  FrSe  A  /\  ta ) )
1918simp1bi 972 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( et  /\  ze ) )
205, 6, 9, 10, 9, 10bnj554 29270 . . 3  |-  ( ( et  /\  ze )  ->  ( ( G `  m )  =  L  <-> 
( G `  suc  i )  =  K ) )
2119, 20syl 16 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( ( G `
 m )  =  L  <->  ( G `  suc  i )  =  K ) )
2215, 21mpbid 202 1  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( G `  suc  i )  =  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    \ cdif 3317    u. cun 3318   (/)c0 3628   {csn 3814   <.cop 3817   U_ciun 4093   suc csuc 4583   omcom 4845    Fn wfn 5449   ` cfv 5454    /\ w-bnj17 29050    predc-bnj14 29052    FrSe w-bnj15 29056
This theorem is referenced by:  bnj571  29277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701  ax-reg 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-eprel 4494  df-id 4498  df-fr 4541  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-bnj17 29051
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