Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1052 Structured version   Visualization version   Unicode version

Theorem bnj1052 29832
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
bnj1052.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1052.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1052.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1052.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1052.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1052.6  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
bnj1052.7  |-  D  =  ( om  \  { (/)
} )
bnj1052.8  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1052.9  |-  ( et  <->  ( ( th  /\  ta  /\ 
ch  /\  ze )  ->  z  e.  B ) )
bnj1052.10  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
bnj1052.37  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
(  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et ) ) )
Assertion
Ref Expression
bnj1052  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Distinct variable groups:    A, f,
i, n, y    z, A, f, i, n    B, f, i, n, z    D, i    R, f, i, n, y    z, R    f, X, i, n, y    z, X    et, j    ta, f,
i, n, z    th, f, i, n, z    i,
j, n    ph, i
Allowed substitution hints:    ph( y, z, f, j, n)    ps( y, z, f, i, j, n)    ch( y, z, f, i, j, n)    th( y,
j)    ta( y, j)    et( y, z, f, i, n)    ze( y, z, f, i, j, n)    rh( y,
z, f, i, j, n)    A( j)    B( y, j)    D( y, z, f, j, n)    R( j)    K( y, z, f, i, j, n)    X( j)

Proof of Theorem bnj1052
StepHypRef Expression
1 bnj1052.1 . 2  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1052.2 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1052.3 . 2  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj1052.4 . 2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
5 bnj1052.5 . 2  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
6 bnj1052.6 . 2  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
7 bnj1052.7 . 2  |-  D  =  ( om  \  { (/)
} )
8 bnj1052.8 . 2  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
9 19.23vv 1829 . . . . 5  |-  ( A. n A. i ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )  <-> 
( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
) )
109albii 1701 . . . 4  |-  ( A. f A. n A. i
( ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  A. f ( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
11 19.23v 1828 . . . 4  |-  ( A. f ( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
1210, 11bitri 257 . . 3  |-  ( A. f A. n A. i
( ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
13 bnj1052.37 . . . . 5  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
(  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et ) ) )
14 vex 3059 . . . . . . . . 9  |-  n  e. 
_V
15 bnj1052.10 . . . . . . . . 9  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
1614, 15bnj110 29717 . . . . . . . 8  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  A. i  e.  n  et )
17 bnj1052.9 . . . . . . . . 9  |-  ( et  <->  ( ( th  /\  ta  /\ 
ch  /\  ze )  ->  z  e.  B ) )
186, 17bnj1049 29831 . . . . . . . 8  |-  ( A. i  e.  n  et  <->  A. i et )
1916, 18sylib 201 . . . . . . 7  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  A. i et )
201919.21bi 1957 . . . . . 6  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  et )
2120, 17sylib 201 . . . . 5  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  ( ( th  /\  ta  /\  ch  /\ 
ze )  ->  z  e.  B ) )
2213, 21mpcom 37 . . . 4  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )
2322gen2 1680 . . 3  |-  A. n A. i ( ( th 
/\  ta  /\  ch  /\  ze )  ->  z  e.  B )
2412, 23mpgbi 1682 . 2  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)
251, 2, 3, 4, 5, 6, 7, 8, 24bnj1034 29827 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991   A.wal 1452    = wceq 1454   E.wex 1673    e. wcel 1897   {cab 2447   A.wral 2748   E.wrex 2749   _Vcvv 3056   [.wsbc 3278    \ cdif 3412    C_ wss 3415   (/)c0 3742   {csn 3979   U_ciun 4291   class class class wbr 4415    _E cep 4761    Fr wfr 4808   suc csuc 5443    Fn wfn 5595   ` cfv 5600   omcom 6718    /\ w-bnj17 29539    predc-bnj14 29541    FrSe w-bnj15 29545    trClc-bnj18 29547    TrFow-bnj19 29549
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  ax-sep 4538
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-iun 4293  df-br 4416  df-fr 4811  df-fn 5603  df-bnj17 29540  df-bnj18 29548
This theorem is referenced by:  bnj1053  29833
  Copyright terms: Public domain W3C validator