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Theorem bnj969 29545
Description: Technical lemma for bnj69 29607. 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 458 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( R  FrSe  A  /\  X  e.  A
) )
2 bnj667 29350 . . . . . . 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 269 . . . . . 6  |-  ( ch 
->  ta )
653ad2ant1 1026 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ta )
76adantl 467 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ta )
83bnj1232 29403 . . . . . . 7  |-  ( ch 
->  n  e.  D
)
9 vex 3090 . . . . . . . 8  |-  m  e. 
_V
109bnj216 29328 . . . . . . 7  |-  ( n  =  suc  m  ->  m  e.  n )
11 id 23 . . . . . . 7  |-  ( p  =  suc  n  ->  p  =  suc  n )
128, 10, 113anim123i 1190 . . . . . 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 991 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1513, 14bitri 252 . . . . . 6  |-  ( si  <->  ( n  e.  D  /\  m  e.  n  /\  p  =  suc  n ) )
1612, 15sylibr 215 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  si )
1716adantl 467 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  si )
181, 7, 17jca32 537 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ta  /\ 
si ) ) )
19 bnj256 29299 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ta  /\  si ) ) )
2018, 19sylibr 215 . 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 29536 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )  e.  _V )
2621, 25syl5eqel 2521 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  C  e.  _V )
2720, 26syl 17 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    \ cdif 3439   (/)c0 3767   {csn 4002   U_ciun 4302   suc csuc 5444    Fn wfn 5596   ` cfv 5601   omcom 6706    /\ w-bnj17 29279    predc-bnj14 29281    FrSe w-bnj15 29285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-bnj17 29280  df-bnj14 29282  df-bnj13 29284  df-bnj15 29286
This theorem is referenced by:  bnj910  29547  bnj1006  29558
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