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Theorem bnj570 29253
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
bnj570.3  |-  D  =  ( om  \  { (/)
} )
bnj570.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj570.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj570.21  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
bnj570.24  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj570.26  |-  G  =  ( f  u.  { <. m ,  C >. } )
bnj570.40  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
bnj570.30  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj570  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  K )
Distinct variable groups:    y, G    y, f    y, i
Allowed substitution hints:    ta( y, f, i, m, n, p)    et( y, f, i, m, n, p)    rh( 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( f, i, m, n, p)    K( y,
f, i, m, n, p)    ph'( y, f, i, m, n, p)    ps'( y, f, i, m, n, p)

Proof of Theorem bnj570
StepHypRef Expression
1 bnj251 29043 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh ) 
<->  ( R  FrSe  A  /\  ( ta  /\  ( et  /\  rh ) ) ) )
2 bnj570.17 . . . . . 6  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
32simp3bi 972 . . . . 5  |-  ( ta 
->  ps' )
4 bnj570.21 . . . . . . . 8  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
54simp1bi 970 . . . . . . 7  |-  ( rh 
->  i  e.  om )
65adantl 452 . . . . . 6  |-  ( ( et  /\  rh )  ->  i  e.  om )
7 bnj570.19 . . . . . . 7  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
87, 4bnj563 29088 . . . . . 6  |-  ( ( et  /\  rh )  ->  suc  i  e.  m )
96, 8jca 518 . . . . 5  |-  ( ( et  /\  rh )  ->  ( i  e. 
om  /\  suc  i  e.  m ) )
10 bnj570.30 . . . . . . . 8  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
1110bnj946 29122 . . . . . . 7  |-  ( ps'  <->  A. i ( i  e. 
om  ->  ( suc  i  e.  m  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
12 sp 1728 . . . . . . 7  |-  ( A. i ( i  e. 
om  ->  ( suc  i  e.  m  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )  ->  (
i  e.  om  ->  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
1311, 12sylbi 187 . . . . . 6  |-  ( ps'  ->  ( i  e.  om  ->  ( suc  i  e.  m  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
1413imp32 422 . . . . 5  |-  ( ( ps'  /\  ( i  e. 
om  /\  suc  i  e.  m ) )  -> 
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
153, 9, 14syl2an 463 . . . 4  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
161, 15bnj833 29104 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
17 bnj570.40 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
1817bnj930 29117 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  Fun  G )
1918bnj721 29102 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  Fun  G )
20 bnj570.26 . . . . . 6  |-  G  =  ( f  u.  { <. m ,  C >. } )
2120bnj931 29118 . . . . 5  |-  f  C_  G
2221a1i 10 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  f  C_  G
)
23 bnj667 29097 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( ta  /\  et  /\  rh ) )
242bnj564 29089 . . . . . . 7  |-  ( ta 
->  dom  f  =  m )
25 eleq2 2357 . . . . . . . 8  |-  ( dom  f  =  m  -> 
( suc  i  e.  dom  f  <->  suc  i  e.  m
) )
2625biimpar 471 . . . . . . 7  |-  ( ( dom  f  =  m  /\  suc  i  e.  m )  ->  suc  i  e.  dom  f )
2724, 8, 26syl2an 463 . . . . . 6  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  suc  i  e.  dom  f )
28273impb 1147 . . . . 5  |-  ( ( ta  /\  et  /\  rh )  ->  suc  i  e.  dom  f )
2923, 28syl 15 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  suc  i  e.  dom  f )
3019, 22, 29bnj1502 29196 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  ( f `  suc  i
) )
312simp1bi 970 . . . . . . . . 9  |-  ( ta 
->  f  Fn  m
)
32 bnj252 29044 . . . . . . . . . . . . . 14  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  <->  ( m  e.  D  /\  (
n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p
) ) )
3332simplbi 446 . . . . . . . . . . . . 13  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  ->  m  e.  D )
347, 33sylbi 187 . . . . . . . . . . . 12  |-  ( et 
->  m  e.  D
)
35 eldifi 3311 . . . . . . . . . . . . 13  |-  ( m  e.  ( om  \  { (/)
} )  ->  m  e.  om )
36 bnj570.3 . . . . . . . . . . . . 13  |-  D  =  ( om  \  { (/)
} )
3735, 36eleq2s 2388 . . . . . . . . . . . 12  |-  ( m  e.  D  ->  m  e.  om )
38 nnord 4680 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  Ord  m )
3934, 37, 383syl 18 . . . . . . . . . . 11  |-  ( et 
->  Ord  m )
4039adantr 451 . . . . . . . . . 10  |-  ( ( et  /\  rh )  ->  Ord  m )
4140, 8jca 518 . . . . . . . . 9  |-  ( ( et  /\  rh )  ->  ( Ord  m  /\  suc  i  e.  m
) )
4231, 41anim12i 549 . . . . . . . 8  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  ( f  Fn  m  /\  ( Ord  m  /\  suc  i  e.  m ) ) )
43 fndm 5359 . . . . . . . . 9  |-  ( f  Fn  m  ->  dom  f  =  m )
44 elelsuc 4480 . . . . . . . . . 10  |-  ( suc  i  e.  m  ->  suc  i  e.  suc  m )
45 ordsucelsuc 4629 . . . . . . . . . . 11  |-  ( Ord  m  ->  ( i  e.  m  <->  suc  i  e.  suc  m ) )
4645biimpar 471 . . . . . . . . . 10  |-  ( ( Ord  m  /\  suc  i  e.  suc  m )  ->  i  e.  m
)
4744, 46sylan2 460 . . . . . . . . 9  |-  ( ( Ord  m  /\  suc  i  e.  m )  ->  i  e.  m )
4843, 47anim12i 549 . . . . . . . 8  |-  ( ( f  Fn  m  /\  ( Ord  m  /\  suc  i  e.  m )
)  ->  ( dom  f  =  m  /\  i  e.  m )
)
49 eleq2 2357 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( i  e.  dom  f 
<->  i  e.  m ) )
5049biimpar 471 . . . . . . . 8  |-  ( ( dom  f  =  m  /\  i  e.  m
)  ->  i  e.  dom  f )
5142, 48, 503syl 18 . . . . . . 7  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  i  e.  dom  f )
52513impb 1147 . . . . . 6  |-  ( ( ta  /\  et  /\  rh )  ->  i  e. 
dom  f )
5323, 52syl 15 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  i  e.  dom  f )
5419, 22, 53bnj1502 29196 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  i )  =  ( f `  i ) )
5554iuneq1d 3944 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
5616, 30, 553eqtr4d 2338 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
57 bnj570.24 . 2  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
5856, 57syl6eqr 2346 1  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  K )
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    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   U_ciun 3921   Ord word 4407   suc csuc 4410   omcom 4672   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj571  29254
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
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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