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Theorem bnj917 29282
Description: Technical lemma for bnj69 29356. 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
bnj917.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj917.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj917.3  |-  D  =  ( om  \  { (/)
} )
bnj917.4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj917.5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
Assertion
Ref Expression
bnj917  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
Distinct variable groups:    A, f,
i, n, y    D, i    R, f, i, n, y    f, X, i, n, y    ph, i
Allowed substitution hints:    ph( y, f, n)    ps( y, f, i, n)    ch( y, f, i, n)    B( y, f, i, n)    D( y, f, n)

Proof of Theorem bnj917
StepHypRef Expression
1 bnj917.1 . . 3  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj917.2 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj917.3 . . 3  |-  D  =  ( om  \  { (/)
} )
4 bnj917.4 . . 3  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
5 biid 227 . . 3  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
61, 2, 3, 4, 5bnj916 29281 . 2  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\ 
ps )  /\  i  e.  n  /\  y  e.  ( f `  i
) ) )
7 bnj917.5 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
8 bnj252 29044 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps ) ) )
97, 8bitri 240 . . . . 5  |-  ( ch  <->  ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
1093anbi1i 1142 . . . 4  |-  ( ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) )  <->  ( (
n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
)  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
11 bnj253 29045 . . . 4  |-  ( ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )  /\  i  e.  n  /\  y  e.  (
f `  i )
)  <->  ( ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps ) )  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
1210, 11bitr4i 243 . . 3  |-  ( ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) )  <->  ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps )  /\  i  e.  n  /\  y  e.  ( f `  i
) ) )
13123exbii 1574 . 2  |-  ( E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) )  <->  E. f E. n E. i ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )  /\  i  e.  n  /\  y  e.  (
f `  i )
) )
146, 13sylibr 203 1  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    \ cdif 3162   (/)c0 3468   {csn 3653   U_ciun 3921   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    trClc-bnj18 29035
This theorem is referenced by:  bnj981  29298  bnj996  29303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-iun 3923  df-fn 5274  df-bnj17 29028  df-bnj18 29036
  Copyright terms: Public domain W3C validator