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Theorem bnj917 31939
Description: Technical lemma for bnj69 32013. 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 236 . . 3  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
61, 2, 3, 4, 5bnj916 31938 . 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 31703 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps ) ) )
97, 8bitri 249 . . . . 5  |-  ( ch  <->  ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
1093anbi1i 1178 . . . 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 31704 . . . 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 252 . . 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 1636 . 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 212 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429   A.wral 2727   E.wrex 2728    \ cdif 3337   (/)c0 3649   {csn 3889   U_ciun 4183   suc csuc 4733    Fn wfn 5425   ` cfv 5430   omcom 6488    /\ w-bnj17 31686    predc-bnj14 31688    trClc-bnj18 31694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-v 2986  df-iun 4185  df-fn 5433  df-bnj17 31687  df-bnj18 31695
This theorem is referenced by:  bnj981  31955  bnj996  31960
  Copyright terms: Public domain W3C validator