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Theorem bnj981 34151
Description: Technical lemma for bnj69 34209. 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
bnj981.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj981.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj981.3  |-  D  =  ( om  \  { (/)
} )
bnj981.4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj981.5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
Assertion
Ref Expression
bnj981  |-  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) )
Distinct variable groups:    A, f,
i, n, y    D, i, y    R, f, i, n, y    f, X, i, n, y    f, Z, i, n, y    ph, i,
y
Allowed substitution hints:    ph( f, n)    ps( y, f, i, n)    ch( y, f, i, n)    B( y, f, i, n)    D( f, n)

Proof of Theorem bnj981
StepHypRef Expression
1 nfv 1708 . . . 4  |-  F/ y  Z  e.  trCl ( X ,  A ,  R )
2 bnj981.2 . . . . . . . . . . . 12  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 nfcv 2619 . . . . . . . . . . . . 13  |-  F/_ y om
4 nfv 1708 . . . . . . . . . . . . . 14  |-  F/ y  suc  i  e.  n
5 nfiu1 4362 . . . . . . . . . . . . . . 15  |-  F/_ y U_ y  e.  (
f `  i )  pred ( y ,  A ,  R )
65nfeq2 2636 . . . . . . . . . . . . . 14  |-  F/ y ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )
74, 6nfim 1921 . . . . . . . . . . . . 13  |-  F/ y ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )
83, 7nfral 2843 . . . . . . . . . . . 12  |-  F/ y A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
92, 8nfxfr 1646 . . . . . . . . . . 11  |-  F/ y ps
109nfri 1875 . . . . . . . . . 10  |-  ( ps 
->  A. y ps )
11 bnj981.5 . . . . . . . . . 10  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
1210, 11bnj1096 33984 . . . . . . . . 9  |-  ( ch 
->  A. y ch )
1312nfi 1624 . . . . . . . 8  |-  F/ y ch
14 nfv 1708 . . . . . . . 8  |-  F/ y  i  e.  n
15 nfv 1708 . . . . . . . 8  |-  F/ y  Z  e.  ( f `
 i )
1613, 14, 15nf3an 1931 . . . . . . 7  |-  F/ y ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) )
1716nfex 1949 . . . . . 6  |-  F/ y E. i ( ch 
/\  i  e.  n  /\  Z  e.  (
f `  i )
)
1817nfex 1949 . . . . 5  |-  F/ y E. n E. i
( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) )
1918nfex 1949 . . . 4  |-  F/ y E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) )
201, 19nfim 1921 . . 3  |-  F/ y ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) ) )
21 eleq1 2529 . . . 4  |-  ( y  =  Z  ->  (
y  e.  trCl ( X ,  A ,  R )  <->  Z  e.  trCl ( X ,  A ,  R ) ) )
22 eleq1 2529 . . . . . 6  |-  ( y  =  Z  ->  (
y  e.  ( f `
 i )  <->  Z  e.  ( f `  i
) ) )
23223anbi3d 1305 . . . . 5  |-  ( y  =  Z  ->  (
( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) )  <->  ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) ) )
24233exbidv 1718 . . . 4  |-  ( y  =  Z  ->  ( E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) )  <->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) ) ) )
2521, 24imbi12d 320 . . 3  |-  ( y  =  Z  ->  (
( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) ) )  <->  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) ) ) )
26 bnj981.1 . . . 4  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
27 bnj981.3 . . . 4  |-  D  =  ( om  \  { (/)
} )
28 bnj981.4 . . . 4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
2926, 2, 27, 28, 11bnj917 34135 . . 3  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
3020, 25, 29vtoclg1f 3166 . 2  |-  ( Z  e.  trCl ( X ,  A ,  R )  ->  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) ) ) )
3130pm2.43i 47 1  |-  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808    \ cdif 3468   (/)c0 3793   {csn 4032   U_ciun 4332   suc csuc 4889    Fn wfn 5589   ` cfv 5594   omcom 6699    /\ w-bnj17 33881    predc-bnj14 33883    trClc-bnj18 33889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-iun 4334  df-fn 5597  df-bnj17 33882  df-bnj18 33890
This theorem is referenced by:  bnj1128  34189
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