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Theorem bnj969 29704
Description: Technical lemma for bnj69 29766. 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 29509 . . . . . . 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 29562 . . . . . . 7  |-  ( ch 
->  n  e.  D
)
9 vex 3020 . . . . . . . 8  |-  m  e. 
_V
109bnj216 29487 . . . . . . 7  |-  ( n  =  suc  m  ->  m  e.  n )
11 id 22 . . . . . . 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 29458 . . 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 29695 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )  e.  _V )
2621, 25syl5eqel 2505 . 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 1872   A.wral 2709   _Vcvv 3017    \ cdif 3371   (/)c0 3699   {csn 3936   U_ciun 4237   suc csuc 5382    Fn wfn 5534   ` cfv 5539   omcom 6645    /\ w-bnj17 29438    predc-bnj14 29440    FrSe w-bnj15 29444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pr 4598  ax-un 6536
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 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-om 6646  df-bnj17 29439  df-bnj14 29441  df-bnj13 29443  df-bnj15 29445
This theorem is referenced by:  bnj910  29706  bnj1006  29717
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