Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj981 Structured version   Visualization version   Unicode version

Theorem bnj981 29809
Description: Technical lemma for bnj69 29867. 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 1771 . . . 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 2602 . . . . . . . . . . . . 13  |-  F/_ y om
4 nfv 1771 . . . . . . . . . . . . . 14  |-  F/ y  suc  i  e.  n
5 nfiu1 4321 . . . . . . . . . . . . . . 15  |-  F/_ y U_ y  e.  (
f `  i )  pred ( y ,  A ,  R )
65nfeq2 2617 . . . . . . . . . . . . . 14  |-  F/ y ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )
74, 6nfim 2013 . . . . . . . . . . . . 13  |-  F/ y ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )
83, 7nfral 2785 . . . . . . . . . . . 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 1706 . . . . . . . . . . 11  |-  F/ y ps
109nfri 1962 . . . . . . . . . 10  |-  ( ps 
->  A. y ps )
11 bnj981.5 . . . . . . . . . 10  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
1210, 11bnj1096 29642 . . . . . . . . 9  |-  ( ch 
->  A. y ch )
1312nfi 1684 . . . . . . . 8  |-  F/ y ch
14 nfv 1771 . . . . . . . 8  |-  F/ y  i  e.  n
15 nfv 1771 . . . . . . . 8  |-  F/ y  Z  e.  ( f `
 i )
1613, 14, 15nf3an 2023 . . . . . . 7  |-  F/ y ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) )
1716nfex 2041 . . . . . 6  |-  F/ y E. i ( ch 
/\  i  e.  n  /\  Z  e.  (
f `  i )
)
1817nfex 2041 . . . . 5  |-  F/ y E. n E. i
( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
) )
1918nfex 2041 . . . 4  |-  F/ y E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) )
201, 19nfim 2013 . . 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 2527 . . . 4  |-  ( y  =  Z  ->  (
y  e.  trCl ( X ,  A ,  R )  <->  Z  e.  trCl ( X ,  A ,  R ) ) )
22 eleq1 2527 . . . . . 6  |-  ( y  =  Z  ->  (
y  e.  ( f `
 i )  <->  Z  e.  ( f `  i
) ) )
23223anbi3d 1354 . . . . 5  |-  ( y  =  Z  ->  (
( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) )  <->  ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) ) )
24233exbidv 1781 . . . 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 326 . . 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 29793 . . 3  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
3020, 25, 29vtoclg1f 3117 . 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 49 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 189    /\ w3a 991    = wceq 1454   E.wex 1673    e. wcel 1897   {cab 2447   A.wral 2748   E.wrex 2749    \ cdif 3412   (/)c0 3742   {csn 3979   U_ciun 4291   suc csuc 5443    Fn wfn 5595   ` cfv 5600   omcom 6718    /\ w-bnj17 29539    predc-bnj14 29541    trClc-bnj18 29547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-rex 2754  df-v 3058  df-iun 4293  df-fn 5603  df-bnj17 29540  df-bnj18 29548
This theorem is referenced by:  bnj1128  29847
  Copyright terms: Public domain W3C validator