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Theorem bnj981 29383
Description: Technical lemma for bnj69 29441. 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 elisset 2968 . . . 4  |-  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. y  y  =  Z )
2 bnj981.1 . . . . . 6  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
3 bnj981.2 . . . . . 6  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
4 bnj981.3 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
5 bnj981.4 . . . . . 6  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
6 bnj981.5 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
72, 3, 4, 5, 6bnj917 29367 . . . . 5  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
8 eleq1 2498 . . . . . 6  |-  ( y  =  Z  ->  (
y  e.  trCl ( X ,  A ,  R )  <->  Z  e.  trCl ( X ,  A ,  R ) ) )
9 eleq1 2498 . . . . . . . 8  |-  ( y  =  Z  ->  (
y  e.  ( f `
 i )  <->  Z  e.  ( f `  i
) ) )
1093anbi3d 1261 . . . . . . 7  |-  ( y  =  Z  ->  (
( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) )  <->  ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) ) )
11103exbidv 1640 . . . . . 6  |-  ( 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
) ) ) )
128, 11imbi12d 313 . . . . 5  |-  ( 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 ) ) ) ) )
137, 12mpbii 204 . . . 4  |-  ( y  =  Z  ->  ( Z  e.  trCl ( X ,  A ,  R
)  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) ) ) )
141, 13bnj593 29175 . . 3  |-  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. y ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) ) )
15 nfv 1630 . . . . 5  |-  F/ y  Z  e.  trCl ( X ,  A ,  R )
16 nfcv 2574 . . . . . . . . . . . . . 14  |-  F/_ y om
17 nfv 1630 . . . . . . . . . . . . . . 15  |-  F/ y  suc  i  e.  n
18 nfiu1 4123 . . . . . . . . . . . . . . . 16  |-  F/_ y U_ y  e.  (
f `  i )  pred ( y ,  A ,  R )
1918nfeq2 2585 . . . . . . . . . . . . . . 15  |-  F/ y ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )
2017, 19nfim 1833 . . . . . . . . . . . . . 14  |-  F/ y ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )
2116, 20nfral 2761 . . . . . . . . . . . . 13  |-  F/ y A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
223, 21nfxfr 1580 . . . . . . . . . . . 12  |-  F/ y ps
2322nfri 1779 . . . . . . . . . . 11  |-  ( ps 
->  A. y ps )
2423, 6bnj1096 29215 . . . . . . . . . 10  |-  ( ch 
->  A. y ch )
2524nfi 1561 . . . . . . . . 9  |-  F/ y ch
26 nfv 1630 . . . . . . . . 9  |-  F/ y  i  e.  n
27 nfv 1630 . . . . . . . . 9  |-  F/ y  Z  e.  ( f `
 i )
2825, 26, 27nf3an 1850 . . . . . . . 8  |-  F/ y ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) )
2928nfex 1866 . . . . . . 7  |-  F/ y E. i ( ch 
/\  i  e.  n  /\  Z  e.  (
f `  i )
)
3029nfex 1866 . . . . . 6  |-  F/ y E. n E. i
( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) )
3130nfex 1866 . . . . 5  |-  F/ y E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) )
3215, 31nfim 1833 . . . 4  |-  F/ y ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) ) )
3332nfri 1779 . . 3  |-  ( ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) ) )  ->  A. y ( Z  e. 
trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) ) )
3414, 33bnj1397 29268 . 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
) ) ) )
3534pm2.43i 46 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708    \ cdif 3319   (/)c0 3630   {csn 3816   U_ciun 4095   suc csuc 4585   omcom 4847    Fn wfn 5451   ` cfv 5456    /\ w-bnj17 29112    predc-bnj14 29114    trClc-bnj18 29120
This theorem is referenced by:  bnj1128  29421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-iun 4097  df-fn 5459  df-bnj17 29113  df-bnj18 29121
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