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Theorem bnj571 29254
Description: Technical lemma for bnj852 29269. 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 1609 . . . 4  |-  F/ i  R  FrSe  A
2 bnj571.17 . . . . 5  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
3 nfv 1609 . . . . . 6  |-  F/ i  f  Fn  m
4 nfv 1609 . . . . . 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 2606 . . . . . . 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 1560 . . . . . 6  |-  F/ i ps'
83, 4, 7nf3an 1786 . . . . 5  |-  F/ i ( f  Fn  m  /\  ph'  /\  ps' )
92, 8nfxfr 1560 . . . 4  |-  F/ i ta
10 nfv 1609 . . . 4  |-  F/ i et
111, 9, 10nf3an 1786 . . 3  |-  F/ i ( R  FrSe  A  /\  ta  /\  et )
12 df-bnj17 29028 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze ) 
<->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze ) )
13 3anass 938 . . . . . . . . . 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 939 . . . . . . . . . 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 936 . . . . . . . . . . . 12  |-  ( ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i )  <->  ( (
i  e.  om  /\  suc  i  e.  n
)  /\  m  =  suc  i ) )
1715, 16bitri 240 . . . . . . . . . . 11  |-  ( ze  <->  ( ( i  e.  om  /\ 
suc  i  e.  n
)  /\  m  =  suc  i ) )
1817anbi2i 675 . . . . . . . . . 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 269 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ze )  <->  ( m  =  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  (
i  e.  om  /\  suc  i  e.  n
) ) )
2012, 19bitri 240 . . . . . . . 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 29250 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  ze )  ->  ( G `  suc  i )  =  K )
3320, 32sylbir 204 . . . . . . 7  |-  ( ( m  =  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  K )
34333expib 1154 . . . . . 6  |-  ( m  =  suc  i  -> 
( ( ( R 
FrSe  A  /\  ta  /\  et )  /\  (
i  e.  om  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  K ) )
35 df-bnj17 29028 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh ) 
<->  ( ( R  FrSe  A  /\  ta  /\  et )  /\  rh ) )
36 3anass 938 . . . . . . . . . 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 939 . . . . . . . . . 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 936 . . . . . . . . . . . 12  |-  ( ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
)  <->  ( ( i  e.  om  /\  suc  i  e.  n )  /\  m  =/=  suc  i
) )
4038, 39bitri 240 . . . . . . . . . . 11  |-  ( rh  <->  ( ( i  e.  om  /\ 
suc  i  e.  n
)  /\  m  =/=  suc  i ) )
4140anbi2i 675 . . . . . . . . . 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 269 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  rh )  <->  ( m  =/=  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  (
i  e.  om  /\  suc  i  e.  n
) ) )
4335, 42bitri 240 . . . . . . . 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 29253 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  K )
4643, 45sylbir 204 . . . . . . 7  |-  ( ( m  =/=  suc  i  /\  ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  K )
47463expib 1154 . . . . . 6  |-  ( m  =/=  suc  i  ->  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  K ) )
4834, 47pm2.61ine 2535 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  et )  /\  ( i  e. 
om  /\  suc  i  e.  n ) )  -> 
( G `  suc  i )  =  K )
4948, 27syl6eq 2344 . . . 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 588 . . 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 1757 . 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 29122 . 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 203 1  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    u. cun 3163   (/)c0 3468   {csn 3653   <.cop 3656   U_ciun 3921   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj600  29267  bnj908  29279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-bnj17 29028
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