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Theorem bnj571 29214
Description: Technical lemma for bnj852 29229. 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
bnj571.3  |-  D  =  ( om  \  { (/)
} )
bnj571.16  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj571.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj571.18  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj571.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj571.20  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
bnj571.22  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
bnj571.23  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
bnj571.24  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj571.25  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
bnj571.26  |-  G  =  ( f  u.  { <. m ,  C >. } )
bnj571.29  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj571.30  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj571.38  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
bnj571.21  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
bnj571.40  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
bnj571.33  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj571  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
Distinct variable groups:    A, i, p, y    y, G    R, i, p, y    et, i    f, i, p, y    i, m, p    i, ph', p
Allowed substitution hints:    ta( x, y, f, i, m, n, p)    et( x, y, f, m, n, p)    ze( x, y, f, i, m, n, p)    si( x, y, f, i, m, n, p)    rh( 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, m, n)    ps'( x, y, f, i, m, n, p)    ps"( x, y, f, i, m, n, p)

Proof of Theorem bnj571
StepHypRef Expression
1 nfv 1629 . . . 4  |-  F/ i  R  FrSe  A
2 bnj571.17 . . . . 5  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
3 nfv 1629 . . . . . 6  |-  F/ i  f  Fn  m
4 nfv 1629 . . . . . 6  |-  F/ i ph'
5 bnj571.30 . . . . . . 7  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
6 nfra1 2748 . . . . . . 7  |-  F/ i A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
75, 6nfxfr 1579 . . . . . 6  |-  F/ i ps'
83, 4, 7nf3an 1849 . . . . 5  |-  F/ i ( f  Fn  m  /\  ph'  /\  ps' )
92, 8nfxfr 1579 . . . 4  |-  F/ i ta
10 nfv 1629 . . . 4  |-  F/ i et
111, 9, 10nf3an 1849 . . 3  |-  F/ i ( R  FrSe  A  /\  ta  /\  et )
12 df-bnj17 28988 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze ) 
<->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze ) )
13 3anass 940 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n )  /\  m  =  suc  i )  <->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  (
( i  e.  om  /\ 
suc  i  e.  n
)  /\  m  =  suc  i ) ) )
14 3anrot 941 . . . . . . . . . 10  |-  ( ( m  =  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  <->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  (
i  e.  om  /\  suc  i  e.  n
)  /\  m  =  suc  i ) )
15 bnj571.20 . . . . . . . . . . . 12  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
16 df-3an 938 . . . . . . . . . . . 12  |-  ( ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i )  <->  ( (
i  e.  om  /\  suc  i  e.  n
)  /\  m  =  suc  i ) )
1715, 16bitri 241 . . . . . . . . . . 11  |-  ( ze  <->  ( ( i  e.  om  /\ 
suc  i  e.  n
)  /\  m  =  suc  i ) )
1817anbi2i 676 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze )  <->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  (
( i  e.  om  /\ 
suc  i  e.  n
)  /\  m  =  suc  i ) ) )
1913, 14, 183bitr4ri 270 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze )  <->  ( m  =  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  (
i  e.  om  /\  suc  i  e.  n
) ) )
2012, 19bitri 241 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze ) 
<->  ( m  =  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) ) )
21 bnj571.3 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
22 bnj571.16 . . . . . . . . 9  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
23 bnj571.18 . . . . . . . . 9  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
24 bnj571.19 . . . . . . . . 9  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
25 bnj571.22 . . . . . . . . 9  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
26 bnj571.23 . . . . . . . . 9  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
27 bnj571.24 . . . . . . . . 9  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
28 bnj571.25 . . . . . . . . 9  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
29 bnj571.26 . . . . . . . . 9  |-  G  =  ( f  u.  { <. m ,  C >. } )
30 bnj571.29 . . . . . . . . 9  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
31 bnj571.38 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
3221, 22, 2, 23, 24, 15, 25, 26, 27, 28, 29, 30, 5, 31bnj558 29210 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( G `  suc  i )  =  K )
3320, 32sylbir 205 . . . . . . 7  |-  ( ( m  =  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  K )
34333expib 1156 . . . . . 6  |-  ( m  =  suc  i  -> 
( ( ( R 
FrSe  A  /\  ta  /\  et )  /\  (
i  e.  om  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  K ) )
35 df-bnj17 28988 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh ) 
<->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  rh ) )
36 3anass 940 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n )  /\  m  =/=  suc  i )  <->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  (
( i  e.  om  /\ 
suc  i  e.  n
)  /\  m  =/=  suc  i ) ) )
37 3anrot 941 . . . . . . . . . 10  |-  ( ( m  =/=  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  <->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  (
i  e.  om  /\  suc  i  e.  n
)  /\  m  =/=  suc  i ) )
38 bnj571.21 . . . . . . . . . . . 12  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
39 df-3an 938 . . . . . . . . . . . 12  |-  ( ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
)  <->  ( ( i  e.  om  /\  suc  i  e.  n )  /\  m  =/=  suc  i
) )
4038, 39bitri 241 . . . . . . . . . . 11  |-  ( rh  <->  ( ( i  e.  om  /\ 
suc  i  e.  n
)  /\  m  =/=  suc  i ) )
4140anbi2i 676 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  rh )  <->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  (
( i  e.  om  /\ 
suc  i  e.  n
)  /\  m  =/=  suc  i ) ) )
4236, 37, 413bitr4ri 270 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  rh )  <->  ( m  =/=  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  (
i  e.  om  /\  suc  i  e.  n
) ) )
4335, 42bitri 241 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh ) 
<->  ( m  =/=  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) ) )
44 bnj571.40 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
4521, 2, 24, 38, 27, 22, 44, 5bnj570 29213 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  K )
4643, 45sylbir 205 . . . . . . 7  |-  ( ( m  =/=  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  K )
47463expib 1156 . . . . . 6  |-  ( m  =/=  suc  i  ->  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  K ) )
4834, 47pm2.61ine 2674 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  K )
4948, 27syl6eq 2483 . . . 4  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
5049exp32 589 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( i  e.  om  ->  ( suc  i  e.  n  ->  ( G `  suc  i )  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) ) )
5111, 50alrimi 1781 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  A. i ( i  e. 
om  ->  ( suc  i  e.  n  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) ) )
52 bnj571.33 . . 3  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
5352bnj946 29082 . 2  |-  ( ps"  <->  A. i
( i  e.  om  ->  ( suc  i  e.  n  ->  ( G `  suc  i )  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) ) )
5451, 53sylibr 204 1  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    \ cdif 3309    u. cun 3310   (/)c0 3620   {csn 3806   <.cop 3809   U_ciun 4085   suc csuc 4575   omcom 4837    Fn wfn 5441   ` cfv 5446    /\ w-bnj17 28987    predc-bnj14 28989    FrSe w-bnj15 28993
This theorem is referenced by:  bnj600  29227  bnj908  29239
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693  ax-reg 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-bnj17 28988
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