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Theorem bnj1030 29791
Description: Technical lemma for bnj69 29814. 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
bnj1030.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1030.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1030.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1030.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1030.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1030.6  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
bnj1030.7  |-  D  =  ( om  \  { (/)
} )
bnj1030.8  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1030.9  |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B ) )
bnj1030.10  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
bnj1030.11  |-  ( ph'  <->  [. j  /  i ]. ph )
bnj1030.12  |-  ( ps'  <->  [. j  /  i ]. ps )
bnj1030.13  |-  ( ch'  <->  [. j  /  i ]. ch )
bnj1030.14  |-  ( th'  <->  [. j  / 
i ]. th )
bnj1030.15  |-  ( ta'  <->  [. j  /  i ]. ta )
bnj1030.16  |-  ( ze'  <->  [. j  /  i ]. ze )
bnj1030.17  |-  ( et'  <->  [. j  /  i ]. et )
bnj1030.18  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
bnj1030.19  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
Assertion
Ref Expression
bnj1030  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Distinct variable groups:    A, f,
i, j, n, y   
z, A, f, i, n    B, f, i, n, y    z, B    D, i, j    R, f, i, j, n, y    z, R    f, X, i, n, y    z, X    ch, j    et, j    ta, f,
i, j, n    th, f,
i, j, n    ph, i    ta, z    th, z
Allowed substitution hints:    ph( y, z, f, j, n)    ps( y, z, f, i, j, n)    ch( y, z, f, i, n)    th( y)    ta( y)    et( y, z, f, i, n)    ze( y,
z, f, i, j, n)    si( y, z, f, i, j, n)    rh( y, z, f, i, j, n)    B( j)    D( y, z, f, n)    K( y, z, f, i, j, n)    X( j)    ph'( y, z, f, i, j, n)    ps'( y, z, f, i, j, n)    ch'( y, z, f, i, j, n)    th'( y, z, f, i, j, n)    ta'( y, z, f, i, j, n)    et'( y, z, f, i, j, n)    ze'( y, z, f, i, j, n)    ph0( y, z, f, i, j, n)

Proof of Theorem bnj1030
StepHypRef Expression
1 bnj1030.1 . 2  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1030.2 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1030.3 . 2  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj1030.4 . 2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
5 bnj1030.5 . 2  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
6 bnj1030.6 . 2  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
7 bnj1030.7 . 2  |-  D  =  ( om  \  { (/)
} )
8 bnj1030.8 . 2  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
9 19.23vv 1808 . . . . 5  |-  ( A. n A. i ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )  <-> 
( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
) )
109albii 1687 . . . 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 1807 . . . 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 252 . . 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 bnj1030.9 . . . . 5  |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B ) )
147bnj1071 29781 . . . . . . . 8  |-  ( n  e.  D  ->  _E  Fr  n )
153, 14bnj769 29568 . . . . . . 7  |-  ( ch 
->  _E  Fr  n )
1615bnj707 29560 . . . . . 6  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  _E  Fr  n )
17 bnj1030.10 . . . . . . 7  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
18 bnj1030.17 . . . . . . 7  |-  ( et'  <->  [. j  /  i ]. et )
19 bnj1030.18 . . . . . . 7  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
20 bnj1030.19 . . . . . . 7  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
212, 8, 13, 18bnj1123 29790 . . . . . . . . . 10  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
222, 3, 5, 7, 19, 20, 21bnj1118 29788 . . . . . . . . 9  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( f `  i
)  C_  B )
231, 3, 5bnj1097 29785 . . . . . . . . 9  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( f `  i )  C_  B
)
2422, 23bnj1109 29593 . . . . . . . 8  |-  E. j
( ( ( th 
/\  ta  /\  ch )  /\  ph0 )  ->  (
f `  i )  C_  B )
2524, 2, 3bnj1093 29784 . . . . . . 7  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i E. j (
ph0  ->  ( f `  i )  C_  B
) )
2613, 17, 18, 19, 20, 25bnj1090 29783 . . . . . 6  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )
27 vex 3084 . . . . . . 7  |-  n  e. 
_V
2827, 17bnj110 29664 . . . . . 6  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  A. i  e.  n  et )
2916, 26, 28syl2anc 665 . . . . 5  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  et )
304, 5, 3, 6, 13, 29, 8bnj1121 29789 . . . 4  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )
3130gen2 1666 . . 3  |-  A. n A. i ( ( th 
/\  ta  /\  ch  /\  ze )  ->  z  e.  B )
3212, 31mpgbi 1668 . 2  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)
331, 2, 3, 4, 5, 6, 7, 8, 32bnj1034 29774 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1868   {cab 2407   A.wral 2775   E.wrex 2776   _Vcvv 3081   [.wsbc 3299    \ cdif 3433    C_ wss 3436   (/)c0 3761   {csn 3996   U_ciun 4296   class class class wbr 4420    _E cep 4758    Fr wfr 4805   dom cdm 4849   suc csuc 5440    Fn wfn 5592   ` cfv 5597   omcom 6702    /\ w-bnj17 29486    predc-bnj14 29488    FrSe w-bnj15 29492    trClc-bnj18 29494    TrFow-bnj19 29496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-tr 4516  df-eprel 4760  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fn 5600  df-fv 5605  df-om 6703  df-bnj17 29487  df-bnj18 29495  df-bnj19 29497
This theorem is referenced by:  bnj1124  29792
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