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Theorem bnj605 28939
Description: Technical lemma. 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
bnj605.5  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
bnj605.13  |-  ( ph"  <->  [. f  / 
f ]. ph )
bnj605.14  |-  ( ps"  <->  [. f  / 
f ]. ps )
bnj605.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj605.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj605.28  |-  f  e. 
_V
bnj605.31  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
bnj605.32  |-  ( ph"  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
bnj605.33  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
bnj605.37  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )
bnj605.38  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )
bnj605.41  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
f  Fn  n )
bnj605.42  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ph" )
bnj605.43  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
Assertion
Ref Expression
bnj605  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
Distinct variable groups:    A, f, m    A, p, f    R, f, m    R, p    et, f    m, n    ph, m    ps, m    x, m    n, p    ph, p    ps, p    th, p    x, p
Allowed substitution hints:    ph( x, y, f, i, n)    ps( x, y, f, i, n)    ch( x, y, f, i, m, n, p)    th( x, y, f, i, m, n)    ta( x, y, f, i, m, n, p)    et( x, y, i, m, n, p)    A( x, y, i, n)    D( x, y, f, i, m, n, p)    R( x, y, i, n)    ph'( x, y, f, i, m, n, p)    ps'( x, y, f, i, m, n, p)    ch'( x, y, f, i, m, n, p)    ph"( x, y, f, i, m, n, p)    ps"( x, y, f, i, m, n, p)

Proof of Theorem bnj605
StepHypRef Expression
1 bnj605.37 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )
21anim1i 551 . . . 4  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  -> 
( E. m E. p et  /\  th )
)
3 nfv 1605 . . . . . . 7  |-  F/ p th
4319.41 1815 . . . . . 6  |-  ( E. p ( et  /\  th )  <->  ( E. p et  /\  th ) )
54exbii 1569 . . . . 5  |-  ( E. m E. p ( et  /\  th )  <->  E. m ( E. p et  /\  th ) )
6 bnj605.5 . . . . . . . 8  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
76bnj1095 28813 . . . . . . 7  |-  ( th 
->  A. m th )
87nfi 1538 . . . . . 6  |-  F/ m th
9819.41 1815 . . . . 5  |-  ( E. m ( E. p et  /\  th )  <->  ( E. m E. p et  /\  th ) )
105, 9bitr2i 241 . . . 4  |-  ( ( E. m E. p et  /\  th )  <->  E. m E. p ( et  /\  th ) )
112, 10sylib 188 . . 3  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  ->  E. m E. p ( et  /\  th )
)
12 bnj605.19 . . . . . . . . . 10  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
1312bnj1232 28836 . . . . . . . . 9  |-  ( et 
->  m  e.  D
)
14 bnj219 28761 . . . . . . . . . 10  |-  ( n  =  suc  m  ->  m  _E  n )
1512, 14bnj770 28793 . . . . . . . . 9  |-  ( et 
->  m  _E  n
)
1613, 15jca 518 . . . . . . . 8  |-  ( et 
->  ( m  e.  D  /\  m  _E  n
) )
1716anim1i 551 . . . . . . 7  |-  ( ( et  /\  th )  ->  ( ( m  e.  D  /\  m  _E  n )  /\  th ) )
18 bnj170 28723 . . . . . . 7  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  <->  ( (
m  e.  D  /\  m  _E  n )  /\  th ) )
1917, 18sylibr 203 . . . . . 6  |-  ( ( et  /\  th )  ->  ( th  /\  m  e.  D  /\  m  _E  n ) )
20 bnj605.38 . . . . . 6  |-  ( ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )
2119, 20syl 15 . . . . 5  |-  ( ( et  /\  th )  ->  ch' )
22 simpl 443 . . . . 5  |-  ( ( et  /\  th )  ->  et )
2321, 22jca 518 . . . 4  |-  ( ( et  /\  th )  ->  ( ch'  /\  et ) )
24232eximi 1564 . . 3  |-  ( E. m E. p ( et  /\  th )  ->  E. m E. p
( ch'  /\  et ) )
25 bnj248 28725 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  <->  ( (
( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  /\  et ) )
26 bnj605.31 . . . . . . . . . . 11  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
27 pm3.35 570 . . . . . . . . . . 11  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) ) )  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) )
2826, 27sylan2b 461 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) )
29 euex 2166 . . . . . . . . . 10  |-  ( E! f ( f  Fn  m  /\  ph'  /\  ps' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
3028, 29syl 15 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
31 bnj605.17 . . . . . . . . 9  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
3230, 31bnj1198 28828 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ta )
3325, 32bnj832 28787 . . . . . . 7  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ta )
34 bnj605.41 . . . . . . . . . . . . . 14  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
f  Fn  n )
35 bnj605.42 . . . . . . . . . . . . . 14  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ph" )
36 bnj605.43 . . . . . . . . . . . . . 14  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
3734, 35, 363jca 1132 . . . . . . . . . . . . 13  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( f  Fn  n  /\  ph"  /\  ps" ) )
38373com23 1157 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  et  /\  ta )  -> 
( f  Fn  n  /\  ph"  /\  ps" ) )
39383expia 1153 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  et )  ->  ( ta 
->  ( f  Fn  n  /\  ph"  /\  ps" ) ) )
4039eximdv 1608 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  et )  ->  ( E. f ta  ->  E. f
( f  Fn  n  /\  ph"  /\  ps" ) ) )
4140adantlr 695 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  et )  ->  ( E. f ta 
->  E. f ( f  Fn  n  /\  ph"  /\  ps" ) ) )
4241adantlr 695 . . . . . . . 8  |-  ( ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  /\  et )  ->  ( E. f ta  ->  E. f
( f  Fn  n  /\  ph"  /\  ps" ) ) )
4325, 42sylbi 187 . . . . . . 7  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  -> 
( E. f ta 
->  E. f ( f  Fn  n  /\  ph"  /\  ps" ) ) )
4433, 43mpd 14 . . . . . 6  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ( f  Fn  n  /\  ph"  /\  ps" ) )
45 bnj432 28741 . . . . . 6  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  <->  ( ( ch' 
/\  et )  /\  ( R  FrSe  A  /\  x  e.  A )
) )
46 biid 227 . . . . . . . 8  |-  ( f  Fn  n  <->  f  Fn  n )
47 bnj605.13 . . . . . . . . 9  |-  ( ph"  <->  [. f  / 
f ]. ph )
48 sbsbc 2995 . . . . . . . . . . 11  |-  ( [ f  /  f ]
ph 
<-> 
[. f  /  f ]. ph )
4948bicomi 193 . . . . . . . . . 10  |-  ( [. f  /  f ]. ph  <->  [ f  /  f ] ph )
50 sbid 1863 . . . . . . . . . 10  |-  ( [ f  /  f ]
ph 
<-> 
ph )
5149, 50bitri 240 . . . . . . . . 9  |-  ( [. f  /  f ]. ph  <->  ph )
5247, 51bitri 240 . . . . . . . 8  |-  ( ph"  <->  ph )
53 bnj605.14 . . . . . . . . 9  |-  ( ps"  <->  [. f  / 
f ]. ps )
54 sbsbc 2995 . . . . . . . . . . 11  |-  ( [ f  /  f ] ps  <->  [. f  /  f ]. ps )
5554bicomi 193 . . . . . . . . . 10  |-  ( [. f  /  f ]. ps  <->  [ f  /  f ] ps )
56 sbid 1863 . . . . . . . . . 10  |-  ( [ f  /  f ] ps  <->  ps )
5755, 56bitri 240 . . . . . . . . 9  |-  ( [. f  /  f ]. ps  <->  ps )
5853, 57bitri 240 . . . . . . . 8  |-  ( ps"  <->  ps )
5946, 52, 583anbi123i 1140 . . . . . . 7  |-  ( ( f  Fn  n  /\  ph"  /\  ps" )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
6059exbii 1569 . . . . . 6  |-  ( E. f ( f  Fn  n  /\  ph"  /\  ps" )  <->  E. f
( f  Fn  n  /\  ph  /\  ps )
)
6144, 45, 603imtr3i 256 . . . . 5  |-  ( ( ( ch'  /\  et )  /\  ( R  FrSe  A  /\  x  e.  A
) )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
)
6261ex 423 . . . 4  |-  ( ( ch'  /\  et )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
6362exlimivv 1667 . . 3  |-  ( E. m E. p ( ch'  /\  et )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
6411, 24, 633syl 18 . 2  |-  ( ( ( n  =/=  1o  /\  n  e.  D )  /\  th )  -> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
65643impa 1146 1  |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  n  /\  ph  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623   [wsb 1629    e. wcel 1684   E!weu 2143    =/= wne 2446   A.wral 2543   _Vcvv 2788   [.wsbc 2991   (/)c0 3455   U_ciun 3905   class class class wbr 4023    _E cep 4303   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255   1oc1o 6472    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717
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-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-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-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-suc 4398  df-bnj17 28712
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