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Theorem bnj570 33259
Description: Technical lemma for bnj852 33275. 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 33051 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh ) 
<->  ( R  FrSe  A  /\  ( ta  /\  ( et  /\  rh ) ) ) )
2 bnj570.17 . . . . . 6  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
32simp3bi 1013 . . . . 5  |-  ( ta 
->  ps' )
4 bnj570.21 . . . . . . . 8  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
54simp1bi 1011 . . . . . . 7  |-  ( rh 
->  i  e.  om )
65adantl 466 . . . . . 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 33096 . . . . . 6  |-  ( ( et  /\  rh )  ->  suc  i  e.  m )
96, 8jca 532 . . . . 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 33129 . . . . . . 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 1808 . . . . . . 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 195 . . . . . 6  |-  ( ps'  ->  ( i  e.  om  ->  ( suc  i  e.  m  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
1413imp32 433 . . . . 5  |-  ( ( ps'  /\  ( i  e. 
om  /\  suc  i  e.  m ) )  -> 
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
153, 9, 14syl2an 477 . . . 4  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
161, 15bnj833 33112 . . 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 33124 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  Fun  G )
1918bnj721 33110 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  Fun  G )
20 bnj570.26 . . . . . 6  |-  G  =  ( f  u.  { <. m ,  C >. } )
2120bnj931 33125 . . . . 5  |-  f  C_  G
2221a1i 11 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  f  C_  G
)
23 bnj667 33105 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( ta  /\  et  /\  rh ) )
242bnj564 33097 . . . . . . 7  |-  ( ta 
->  dom  f  =  m )
25 eleq2 2540 . . . . . . . 8  |-  ( dom  f  =  m  -> 
( suc  i  e.  dom  f  <->  suc  i  e.  m
) )
2625biimpar 485 . . . . . . 7  |-  ( ( dom  f  =  m  /\  suc  i  e.  m )  ->  suc  i  e.  dom  f )
2724, 8, 26syl2an 477 . . . . . 6  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  suc  i  e.  dom  f )
28273impb 1192 . . . . 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 33202 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  ( f `  suc  i
) )
312simp1bi 1011 . . . . . . . . 9  |-  ( ta 
->  f  Fn  m
)
32 bnj252 33052 . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . 13  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  ->  m  e.  D )
347, 33sylbi 195 . . . . . . . . . . . 12  |-  ( et 
->  m  e.  D
)
35 eldifi 3626 . . . . . . . . . . . . 13  |-  ( m  e.  ( om  \  { (/)
} )  ->  m  e.  om )
36 bnj570.3 . . . . . . . . . . . . 13  |-  D  =  ( om  \  { (/)
} )
3735, 36eleq2s 2575 . . . . . . . . . . . 12  |-  ( m  e.  D  ->  m  e.  om )
38 nnord 6693 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  Ord  m )
3934, 37, 383syl 20 . . . . . . . . . . 11  |-  ( et 
->  Ord  m )
4039adantr 465 . . . . . . . . . 10  |-  ( ( et  /\  rh )  ->  Ord  m )
4140, 8jca 532 . . . . . . . . 9  |-  ( ( et  /\  rh )  ->  ( Ord  m  /\  suc  i  e.  m
) )
4231, 41anim12i 566 . . . . . . . 8  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  ( f  Fn  m  /\  ( Ord  m  /\  suc  i  e.  m ) ) )
43 fndm 5680 . . . . . . . . 9  |-  ( f  Fn  m  ->  dom  f  =  m )
44 elelsuc 4950 . . . . . . . . . 10  |-  ( suc  i  e.  m  ->  suc  i  e.  suc  m )
45 ordsucelsuc 6642 . . . . . . . . . . 11  |-  ( Ord  m  ->  ( i  e.  m  <->  suc  i  e.  suc  m ) )
4645biimpar 485 . . . . . . . . . 10  |-  ( ( Ord  m  /\  suc  i  e.  suc  m )  ->  i  e.  m
)
4744, 46sylan2 474 . . . . . . . . 9  |-  ( ( Ord  m  /\  suc  i  e.  m )  ->  i  e.  m )
4843, 47anim12i 566 . . . . . . . 8  |-  ( ( f  Fn  m  /\  ( Ord  m  /\  suc  i  e.  m )
)  ->  ( dom  f  =  m  /\  i  e.  m )
)
49 eleq2 2540 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( i  e.  dom  f 
<->  i  e.  m ) )
5049biimpar 485 . . . . . . . 8  |-  ( ( dom  f  =  m  /\  i  e.  m
)  ->  i  e.  dom  f )
5142, 48, 503syl 20 . . . . . . 7  |-  ( ( ta  /\  ( et 
/\  rh ) )  ->  i  e.  dom  f )
52513impb 1192 . . . . . 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 33202 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  i )  =  ( f `  i ) )
5554iuneq1d 4350 . . 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 2518 . 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 2526 1  |-  ( ( R  FrSe  A  /\  ta  /\  et  /\  rh )  ->  ( G `  suc  i )  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U_ciun 4325   Ord word 4877   suc csuc 4880   dom cdm 4999   Fun wfun 5582    Fn wfn 5583   ` cfv 5588   omcom 6685    /\ w-bnj17 33035    predc-bnj14 33037    FrSe w-bnj15 33041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-om 6686  df-bnj17 33036
This theorem is referenced by:  bnj571  33260
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