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Theorem bnj969 28978
Description: Technical lemma for bnj69 29040. 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
bnj969.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj969.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj969.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj969.10  |-  D  =  ( om  \  { (/)
} )
bnj969.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj969.14  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj969.15  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
Assertion
Ref Expression
bnj969  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
Distinct variable groups:    A, i, m, y    R, i, m, y    f, i, m, y    i, n, m
Allowed substitution hints:    ph( y, f, i, m, n, p)    ps( y, f, i, m, n, p)    ch( y,
f, i, m, n, p)    ta( y, f, i, m, n, p)    si( y,
f, i, m, n, p)    A( f, n, p)    C( y, f, i, m, n, p)    D( y,
f, i, m, n, p)    R( f, n, p)    X( y, f, i, m, n, p)

Proof of Theorem bnj969
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( R  FrSe  A  /\  X  e.  A
) )
2 bnj667 28781 . . . . . . 7  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
f  Fn  n  /\  ph 
/\  ps ) )
3 bnj969.3 . . . . . . 7  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj969.14 . . . . . . 7  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
52, 3, 43imtr4i 257 . . . . . 6  |-  ( ch 
->  ta )
653ad2ant1 976 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ta )
76adantl 452 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ta )
83bnj1232 28836 . . . . . . 7  |-  ( ch 
->  n  e.  D
)
9 vex 2791 . . . . . . . 8  |-  m  e. 
_V
109bnj216 28760 . . . . . . 7  |-  ( n  =  suc  m  ->  m  e.  n )
11 id 19 . . . . . . 7  |-  ( p  =  suc  n  ->  p  =  suc  n )
128, 10, 113anim123i 1137 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
13 bnj969.15 . . . . . . 7  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
14 3ancomb 943 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1513, 14bitri 240 . . . . . 6  |-  ( si  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1612, 15sylibr 203 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  si )
1716adantl 452 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  si )
181, 7, 17jca32 521 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ta  /\ 
si ) ) )
19 bnj256 28731 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ta  /\  si ) ) )
2018, 19sylibr 203 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )
)
21 bnj969.12 . . 3  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
22 bnj969.10 . . . 4  |-  D  =  ( om  \  { (/)
} )
23 bnj969.1 . . . 4  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
24 bnj969.2 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2522, 4, 13, 23, 24bnj938 28969 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )  e.  _V )
2621, 25syl5eqel 2367 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  C  e.  _V )
2720, 26syl 15 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149   (/)c0 3455   {csn 3640   U_ciun 3905   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj910  28980  bnj1006  28991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718
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