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Theorem bnj1052 34451
Description: Technical lemma for bnj69 34486. 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
bnj1052.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1052.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1052.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1052.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1052.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1052.6  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
bnj1052.7  |-  D  =  ( om  \  { (/)
} )
bnj1052.8  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1052.9  |-  ( et  <->  ( ( th  /\  ta  /\ 
ch  /\  ze )  ->  z  e.  B ) )
bnj1052.10  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
bnj1052.37  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
(  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et ) ) )
Assertion
Ref Expression
bnj1052  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Distinct variable groups:    A, f,
i, n, y    z, A, f, i, n    B, f, i, n, z    D, i    R, f, i, n, y    z, R    f, X, i, n, y    z, X    et, j    ta, f,
i, n, z    th, f, i, n, z    i,
j, n    ph, i
Allowed substitution hints:    ph( y, z, f, j, n)    ps( y, z, f, i, j, n)    ch( y, z, f, i, j, n)    th( y,
j)    ta( y, j)    et( y, z, f, i, n)    ze( y, z, f, i, j, n)    rh( y,
z, f, i, j, n)    A( j)    B( y, j)    D( y, z, f, j, n)    R( j)    K( y, z, f, i, j, n)    X( j)

Proof of Theorem bnj1052
StepHypRef Expression
1 bnj1052.1 . 2  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1052.2 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1052.3 . 2  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj1052.4 . 2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
5 bnj1052.5 . 2  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
6 bnj1052.6 . 2  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
7 bnj1052.7 . 2  |-  D  =  ( om  \  { (/)
} )
8 bnj1052.8 . 2  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
9 19.23vv 1766 . . . . 5  |-  ( A. n A. i ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )  <-> 
( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
) )
109albii 1645 . . . 4  |-  ( A. f A. n A. i
( ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  A. f ( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
11 19.23v 1765 . . . 4  |-  ( A. f ( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
1210, 11bitri 249 . . 3  |-  ( A. f A. n A. i
( ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
13 bnj1052.37 . . . . 5  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
(  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et ) ) )
14 vex 3109 . . . . . . . . 9  |-  n  e. 
_V
15 bnj1052.10 . . . . . . . . 9  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
1614, 15bnj110 34336 . . . . . . . 8  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  A. i  e.  n  et )
17 bnj1052.9 . . . . . . . . 9  |-  ( et  <->  ( ( th  /\  ta  /\ 
ch  /\  ze )  ->  z  e.  B ) )
186, 17bnj1049 34450 . . . . . . . 8  |-  ( A. i  e.  n  et  <->  A. i et )
1916, 18sylib 196 . . . . . . 7  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  A. i et )
201919.21bi 1874 . . . . . 6  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  et )
2120, 17sylib 196 . . . . 5  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  ( ( th  /\  ta  /\  ch  /\ 
ze )  ->  z  e.  B ) )
2213, 21mpcom 36 . . . 4  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )
2322gen2 1624 . . 3  |-  A. n A. i ( ( th 
/\  ta  /\  ch  /\  ze )  ->  z  e.  B )
2412, 23mpgbi 1626 . 2  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)
251, 2, 3, 4, 5, 6, 7, 8, 24bnj1034 34446 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805   _Vcvv 3106   [.wsbc 3324    \ cdif 3458    C_ wss 3461   (/)c0 3783   {csn 4016   U_ciun 4315   class class class wbr 4439    _E cep 4778    Fr wfr 4824   suc csuc 4869    Fn wfn 5565   ` cfv 5570   omcom 6673    /\ w-bnj17 34158    predc-bnj14 34160    FrSe w-bnj15 34164    trClc-bnj18 34166    TrFow-bnj19 34168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-iun 4317  df-br 4440  df-fr 4827  df-fn 5573  df-bnj17 34159  df-bnj18 34167
This theorem is referenced by:  bnj1053  34452
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