Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj967 Structured version   Unicode version

Theorem bnj967 29316
Description: Technical lemma for bnj69 29379. 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
bnj967.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj967.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj967.10  |-  D  =  ( om  \  { (/)
} )
bnj967.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj967.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj967.44  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
Assertion
Ref Expression
bnj967  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
Distinct variable groups:    y, f    y, i    y, n
Allowed substitution hints:    ph( y, f, i, m, n, p)    ps( y, f, i, m, n, p)    ch( y,
f, i, m, n, p)    A( y, f, i, m, n, p)    C( y, f, i, m, n, p)    D( y, f, i, m, n, p)    R( y, f, i, m, n, p)    G( y, f, i, m, n, p)    X( y, f, i, m, n, p)

Proof of Theorem bnj967
StepHypRef Expression
1 bnj967.44 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
213adant3 977 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  C  e.  _V )
3 bnj967.3 . . . . . . . . 9  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
43bnj1235 29176 . . . . . . . 8  |-  ( ch 
->  f  Fn  n
)
543ad2ant1 978 . . . . . . 7  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
653ad2ant2 979 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  f  Fn  n )
7 simp23 992 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  p  =  suc  n )
8 simp3 959 . . . . . . 7  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
)  ->  suc  i  e.  n )
983ad2ant3 980 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  suc  i  e.  n )
102, 6, 7, 9bnj951 29146 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n )
)
11 bnj967.10 . . . . . . . . . 10  |-  D  =  ( om  \  { (/)
} )
1211bnj923 29137 . . . . . . . . 9  |-  ( n  e.  D  ->  n  e.  om )
133, 12bnj769 29131 . . . . . . . 8  |-  ( ch 
->  n  e.  om )
14133ad2ant1 978 . . . . . . 7  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  om )
1514, 8bnj240 29063 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  (
n  e.  om  /\  suc  i  e.  n
) )
16 nnord 4853 . . . . . . . 8  |-  ( n  e.  om  ->  Ord  n )
17 ordtr 4595 . . . . . . . 8  |-  ( Ord  n  ->  Tr  n
)
1816, 17syl 16 . . . . . . 7  |-  ( n  e.  om  ->  Tr  n )
19 trsuc 4666 . . . . . . 7  |-  ( ( Tr  n  /\  suc  i  e.  n )  ->  i  e.  n )
2018, 19sylan 458 . . . . . 6  |-  ( ( n  e.  om  /\  suc  i  e.  n
)  ->  i  e.  n )
2115, 20syl 16 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  i  e.  n )
22 bnj658 29119 . . . . . . 7  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n
)  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n ) )
2322anim1i 552 . . . . . 6  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n
)  /\  i  e.  n )  ->  (
( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  /\  i  e.  n
) )
24 df-bnj17 29051 . . . . . 6  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )  <->  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  /\  i  e.  n
) )
2523, 24sylibr 204 . . . . 5  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n
)  /\  i  e.  n )  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n ) )
2610, 21, 25syl2anc 643 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n ) )
27 bnj967.13 . . . . 5  |-  G  =  ( f  u.  { <. n ,  C >. } )
2827bnj945 29144 . . . 4  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )  ->  ( G `  i
)  =  ( f `
 i ) )
2926, 28syl 16 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( G `  i )  =  ( f `  i ) )
3027bnj945 29144 . . . 4  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n
)  ->  ( G `  suc  i )  =  ( f `  suc  i ) )
3110, 30syl 16 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  ( f `  suc  i ) )
32 3simpb 955 . . . 4  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
)  ->  ( i  e.  om  /\  suc  i  e.  n ) )
33323ad2ant3 980 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  (
i  e.  om  /\  suc  i  e.  n
) )
343bnj1254 29181 . . . . 5  |-  ( ch 
->  ps )
35343ad2ant1 978 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ps )
36353ad2ant2 979 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ps )
3729, 31, 33, 36bnj951 29146 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  (
( G `  i
)  =  ( f `
 i )  /\  ( G `  suc  i
)  =  ( f `
 suc  i )  /\  ( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )
)
38 bnj967.2 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
39 bnj967.12 . . . 4  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
4039, 27bnj958 29311 . . 3  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
4138, 40bnj953 29310 . 2  |-  ( ( ( G `  i
)  =  ( f `
 i )  /\  ( G `  suc  i
)  =  ( f `
 suc  i )  /\  ( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
4237, 41syl 16 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    \ cdif 3317    u. cun 3318   (/)c0 3628   {csn 3814   <.cop 3817   U_ciun 4093   Tr wtr 4302   Ord word 4580   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:  bnj910  29319
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-3or 937  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-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  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
  Copyright terms: Public domain W3C validator