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Theorem bnj917 28966
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
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 28965 . 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 28728 . . . . . 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 28729 . . . 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 1571 . 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 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    \ cdif 3149   (/)c0 3455   {csn 3640   U_ciun 3905   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    trClc-bnj18 28719
This theorem is referenced by:  bnj981  28982  bnj996  28987
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-iun 3907  df-fn 5258  df-bnj17 28712  df-bnj18 28720
  Copyright terms: Public domain W3C validator