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Theorem bnj570 29277
Description: Technical lemma for bnj852 29293. 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 29067 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh ) 
<->  ( R  FrSe  A  /\  ( ta  /\  ( et  /\  rh ) ) ) )
2 bnj570.17 . . . . . 6  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
32simp3bi 975 . . . . 5  |-  ( ta 
->  ps' )
4 bnj570.21 . . . . . . . 8  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
54simp1bi 973 . . . . . . 7  |-  ( rh 
->  i  e.  om )
65adantl 454 . . . . . 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 29112 . . . . . 6  |-  ( ( et  /\  rh )  ->  suc  i  e.  m )
96, 8jca 520 . . . . 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 29146 . . . . . . 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 1764 . . . . . . 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 189 . . . . . 6  |-  ( ps'  ->  ( i  e.  om  ->  ( suc  i  e.  m  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
1413imp32 424 . . . . 5  |-  ( ( ps'  /\  ( i  e. 
om  /\  suc  i  e.  m ) )  -> 
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
153, 9, 14syl2an 465 . . . 4  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
161, 15bnj833 29128 . . 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 29141 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  Fun  G )
1918bnj721 29126 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  Fun  G )
20 bnj570.26 . . . . . 6  |-  G  =  ( f  u.  { <. m ,  C >. } )
2120bnj931 29142 . . . . 5  |-  f  C_  G
2221a1i 11 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  f  C_  G
)
23 bnj667 29121 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( ta  /\  et  /\  rh ) )
242bnj564 29113 . . . . . . 7  |-  ( ta 
->  dom  f  =  m )
25 eleq2 2498 . . . . . . . 8  |-  ( dom  f  =  m  -> 
( suc  i  e.  dom  f  <->  suc  i  e.  m
) )
2625biimpar 473 . . . . . . 7  |-  ( ( dom  f  =  m  /\  suc  i  e.  m )  ->  suc  i  e.  dom  f )
2724, 8, 26syl2an 465 . . . . . 6  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  suc  i  e.  dom  f )
28273impb 1150 . . . . 5  |-  ( ( ta  /\  et  /\  rh )  ->  suc  i  e.  dom  f )
2923, 28syl 16 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  suc  i  e.  dom  f )
3019, 22, 29bnj1502 29220 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  ( f `  suc  i
) )
312simp1bi 973 . . . . . . . . 9  |-  ( ta 
->  f  Fn  m
)
32 bnj252 29068 . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . 13  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  ->  m  e.  D )
347, 33sylbi 189 . . . . . . . . . . . 12  |-  ( et 
->  m  e.  D
)
35 eldifi 3470 . . . . . . . . . . . . 13  |-  ( m  e.  ( om  \  { (/)
} )  ->  m  e.  om )
36 bnj570.3 . . . . . . . . . . . . 13  |-  D  =  ( om  \  { (/)
} )
3735, 36eleq2s 2529 . . . . . . . . . . . 12  |-  ( m  e.  D  ->  m  e.  om )
38 nnord 4854 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  Ord  m )
3934, 37, 383syl 19 . . . . . . . . . . 11  |-  ( et 
->  Ord  m )
4039adantr 453 . . . . . . . . . 10  |-  ( ( et  /\  rh )  ->  Ord  m )
4140, 8jca 520 . . . . . . . . 9  |-  ( ( et  /\  rh )  ->  ( Ord  m  /\  suc  i  e.  m
) )
4231, 41anim12i 551 . . . . . . . 8  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  ( f  Fn  m  /\  ( Ord  m  /\  suc  i  e.  m ) ) )
43 fndm 5545 . . . . . . . . 9  |-  ( f  Fn  m  ->  dom  f  =  m )
44 elelsuc 4654 . . . . . . . . . 10  |-  ( suc  i  e.  m  ->  suc  i  e.  suc  m )
45 ordsucelsuc 4803 . . . . . . . . . . 11  |-  ( Ord  m  ->  ( i  e.  m  <->  suc  i  e.  suc  m ) )
4645biimpar 473 . . . . . . . . . 10  |-  ( ( Ord  m  /\  suc  i  e.  suc  m )  ->  i  e.  m
)
4744, 46sylan2 462 . . . . . . . . 9  |-  ( ( Ord  m  /\  suc  i  e.  m )  ->  i  e.  m )
4843, 47anim12i 551 . . . . . . . 8  |-  ( ( f  Fn  m  /\  ( Ord  m  /\  suc  i  e.  m )
)  ->  ( dom  f  =  m  /\  i  e.  m )
)
49 eleq2 2498 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( i  e.  dom  f 
<->  i  e.  m ) )
5049biimpar 473 . . . . . . . 8  |-  ( ( dom  f  =  m  /\  i  e.  m
)  ->  i  e.  dom  f )
5142, 48, 503syl 19 . . . . . . 7  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  i  e.  dom  f )
52513impb 1150 . . . . . 6  |-  ( ( ta  /\  et  /\  rh )  ->  i  e. 
dom  f )
5323, 52syl 16 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  i  e.  dom  f )
5419, 22, 53bnj1502 29220 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  i )  =  ( f `  i ) )
5554iuneq1d 4117 . . 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 2479 . 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 2487 1  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706    \ cdif 3318    u. cun 3319    C_ wss 3321   (/)c0 3629   {csn 3815   <.cop 3818   U_ciun 4094   Ord word 4581   suc csuc 4584   omcom 4846   dom cdm 4879   Fun wfun 5449    Fn wfn 5450   ` cfv 5455    /\ w-bnj17 29051    predc-bnj14 29053    FrSe w-bnj15 29057
This theorem is referenced by:  bnj571  29278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-res 4891  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463  df-bnj17 29052
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