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Theorem bnj1034 29000
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
bnj1034.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1034.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1034.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1034.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1034.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1034.7  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
bnj1034.8  |-  D  =  ( om  \  { (/)
} )
bnj1034.9  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1034.10  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)
Assertion
Ref Expression
bnj1034  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Distinct variable groups:    A, f,
i, n, y    z, A, f, i, n    z, B    D, i    R, f, i, n, y    z, R    f, X, i, n, y    z, X    ta, f, i, n, z    th, f, i, n, z    ph, i
Allowed substitution hints:    ph( y, z, f, n)    ps( y,
z, f, i, n)    ch( y, z, f, i, n)    th( y)    ta( y)    ze( y, z, f, i, n)    B( y, f, i, n)    D( y, z, f, n)    K( y, z, f, i, n)

Proof of Theorem bnj1034
StepHypRef Expression
1 bnj1034.1 . 2  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1034.2 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1034.3 . 2  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj1034.4 . 2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
5 bnj1034.5 . 2  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
6 biid 227 . 2  |-  ( z  e.  trCl ( X ,  A ,  R )  <->  z  e.  trCl ( X ,  A ,  R )
)
7 bnj1034.7 . 2  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
8 bnj1034.8 . 2  |-  D  =  ( om  \  { (/)
} )
9 bnj1034.9 . 2  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
10 bnj1034.10 . 2  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10bnj1033 28999 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)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    trClc-bnj18 28719    TrFow-bnj19 28721
This theorem is referenced by:  bnj1052  29005  bnj1030  29017
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iun 3907  df-fn 5258  df-bnj17 28712  df-bnj18 28720
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