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Theorem bnj151 28909
Description: Technical lemma for bnj153 28912. 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
bnj151.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj151.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj151.3  |-  D  =  ( om  \  { (/)
} )
bnj151.4  |-  ( th  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
bnj151.5  |-  ( ta  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. th ) )
bnj151.6  |-  ( ze  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
bnj151.7  |-  ( ph'  <->  [. 1o  /  n ]. ph )
bnj151.8  |-  ( ps'  <->  [. 1o  /  n ]. ps )
bnj151.9  |-  ( th'  <->  [. 1o  /  n ]. th )
bnj151.10  |-  ( th0  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
bnj151.11  |-  ( th1  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E* f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
bnj151.12  |-  ( ze'  <->  [. 1o  /  n ]. ze )
bnj151.13  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
bnj151.14  |-  ( ph"  <->  [. F  / 
f ]. ph' )
bnj151.15  |-  ( ps"  <->  [. F  / 
f ]. ps' )
bnj151.16  |-  ( ze"  <->  [. F  / 
f ]. ze' )
bnj151.17  |-  ( ze0  <->  (
f  Fn  1o  /\  ph' 
/\  ps' ) )
bnj151.18  |-  ( ze1  <->  [. g  /  f ]. ze0 )
bnj151.19  |-  ( ph1  <->  [. g  /  f ]. ph' )
bnj151.20  |-  ( ps1  <->  [. g  /  f ]. ps' )
Assertion
Ref Expression
bnj151  |-  ( n  =  1o  ->  (
( n  e.  D  /\  ta )  ->  th )
)
Distinct variable groups:    A, f,
g, x    A, n, f, x    f, F, i, y    R, f, g, x    R, n    f, ze1    f, ze"    g, ze0    i, n, y    m, n
Allowed substitution hints:    ph( x, y, f, g, i, m, n)    ps( x, y, f, g, i, m, n)    th( x, y, f, g, i, m, n)    ta( x, y, f, g, i, m, n)    ze( x, y, f, g, i, m, n)    A( y, i, m)    D( x, y, f, g, i, m, n)    R( y, i, m)    F( x, g, m, n)    ph'( x, y, f, g, i, m, n)    ps'( x, y, f, g, i, m, n)    th'( x, y, f, g, i, m, n)    ze'( x, y, f, g, i, m, n)   
ph"( x, y, f, g, i, m, n)    ps"( x, y, f, g, i, m, n)    ze"( x, y, g, i, m, n)    th0( x, y, f, g, i, m, n)    ze0( x, y, f, i, m, n)    ph1( x, y, f, g, i, m, n)    ps1( x, y, f, g, i, m, n)    th1( x, y, f, g, i, m, n)    ze1( x, y, g, i, m, n)

Proof of Theorem bnj151
StepHypRef Expression
1 bnj151.1 . . . . . . 7  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
2 bnj151.2 . . . . . . 7  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj151.6 . . . . . . 7  |-  ( ze  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
4 bnj151.7 . . . . . . 7  |-  ( ph'  <->  [. 1o  /  n ]. ph )
5 bnj151.8 . . . . . . 7  |-  ( ps'  <->  [. 1o  /  n ]. ps )
6 bnj151.10 . . . . . . 7  |-  ( th0  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
7 bnj151.12 . . . . . . 7  |-  ( ze'  <->  [. 1o  /  n ]. ze )
8 bnj151.13 . . . . . . 7  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
9 bnj151.14 . . . . . . 7  |-  ( ph"  <->  [. F  / 
f ]. ph' )
10 bnj151.15 . . . . . . 7  |-  ( ps"  <->  [. F  / 
f ]. ps' )
11 bnj151.16 . . . . . . 7  |-  ( ze"  <->  [. F  / 
f ]. ze' )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj150 28908 . . . . . 6  |-  th0
1312, 6mpbi 199 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f ( f  Fn  1o  /\  ph'  /\  ps' ) )
14 bnj151.11 . . . . . . 7  |-  ( th1  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E* f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
15 bnj151.17 . . . . . . 7  |-  ( ze0  <->  (
f  Fn  1o  /\  ph' 
/\  ps' ) )
16 bnj151.18 . . . . . . 7  |-  ( ze1  <->  [. g  /  f ]. ze0 )
17 bnj151.19 . . . . . . 7  |-  ( ph1  <->  [. g  /  f ]. ph' )
18 bnj151.20 . . . . . . 7  |-  ( ps1  <->  [. g  /  f ]. ps' )
191, 4bnj118 28901 . . . . . . 7  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
2014, 15, 16, 17, 18, 19bnj149 28907 . . . . . 6  |-  th1
2120, 14mpbi 199 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) )
22 eu5 2181 . . . . 5  |-  ( E! f ( f  Fn  1o  /\  ph'  /\  ps' )  <->  ( E. f ( f  Fn  1o  /\  ph'  /\  ps' )  /\  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
2313, 21, 22sylanbrc 645 . . . 4  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  1o  /\  ph'  /\  ps' ) )
24 bnj151.4 . . . . 5  |-  ( th  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
25 bnj151.9 . . . . 5  |-  ( th'  <->  [. 1o  /  n ]. th )
2624, 4, 5, 25bnj130 28906 . . . 4  |-  ( th'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
2723, 26mpbir 200 . . 3  |-  th'
28 sbceq1a 3001 . . . 4  |-  ( n  =  1o  ->  ( th 
<-> 
[. 1o  /  n ]. th ) )
2928, 25syl6bbr 254 . . 3  |-  ( n  =  1o  ->  ( th 
<->  th' ) )
3027, 29mpbiri 224 . 2  |-  ( n  =  1o  ->  th )
3130a1d 22 1  |-  ( n  =  1o  ->  (
( n  e.  D  /\  ta )  ->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   E*wmo 2144   A.wral 2543   [.wsbc 2991    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   U_ciun 3905   class class class wbr 4023    _E cep 4303   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255   1oc1o 6472    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj153  28912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj13 28716  df-bnj15 28718
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