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Theorem bnj908 28474
Description: Technical lemma for bnj69 28551. 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
bnj908.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj908.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj908.3  |-  D  =  ( om  \  { (/)
} )
bnj908.4  |-  ( ch  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
bnj908.5  |-  ( th  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. ch ) )
bnj908.10  |-  ( ph'  <->  [. m  /  n ]. ph )
bnj908.11  |-  ( ps'  <->  [. m  /  n ]. ps )
bnj908.12  |-  ( ch'  <->  [. m  /  n ]. ch )
bnj908.13  |-  ( ph"  <->  [. G  / 
f ]. ph )
bnj908.14  |-  ( ps"  <->  [. G  / 
f ]. ps )
bnj908.15  |-  ( ch"  <->  [. G  / 
f ]. ch )
bnj908.16  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj908.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj908.18  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj908.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj908.20  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
bnj908.21  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
bnj908.22  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
bnj908.23  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
bnj908.24  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj908.25  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
bnj908.26  |-  G  =  ( f  u.  { <. m ,  C >. } )
Assertion
Ref Expression
bnj908  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ( G  Fn  n  /\  ph"  /\  ps" ) )
Distinct variable groups:    A, f,
i, m, n, p   
y, A, f, i, n, p    D, p   
i, G, y    R, f, i, m, n, p   
y, R    et, f,
i    x, f, m, n, p    i, ph', p    ph, m, p    ps, m, p    th, 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, m, n, p)    ze( x, y, f, i, m, n, p)    si( x, y, f, i, m, n, p)    rh( x, y, f, i, m, n, p)    A( x)    B( x, y, f, i, m, n, p)    C( x, y, f, i, m, n, p)    D( x, y, f, i, m, n)    R( x)    G( x, f, m, n, p)    K( x, y, f, i, m, n, p)    L( x, y, f, i, m, n, p)    ph'( x, y, f, m, n)    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)    ch"( x, y, f, i, m, n, p)

Proof of Theorem bnj908
StepHypRef Expression
1 bnj248 28236 . . . . . 6  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  <->  ( (
( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  /\  et ) )
2 bnj908.4 . . . . . . . . . . 11  |-  ( ch  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
3 bnj908.10 . . . . . . . . . . 11  |-  ( ph'  <->  [. m  /  n ]. ph )
4 bnj908.11 . . . . . . . . . . 11  |-  ( ps'  <->  [. m  /  n ]. ps )
5 bnj908.12 . . . . . . . . . . 11  |-  ( ch'  <->  [. m  /  n ]. ch )
6 vex 2825 . . . . . . . . . . 11  |-  m  e. 
_V
72, 3, 4, 5, 6bnj207 28424 . . . . . . . . . 10  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
87biimpi 186 . . . . . . . . 9  |-  ( ch'  ->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
9 euex 2199 . . . . . . . . 9  |-  ( E! f ( f  Fn  m  /\  ph'  /\  ps' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
108, 9syl6 29 . . . . . . . 8  |-  ( ch'  ->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  m  /\  ph'  /\  ps' ) ) )
1110impcom 419 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ( f  Fn  m  /\  ph'  /\  ps' ) )
12 bnj908.17 . . . . . . 7  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
1311, 12bnj1198 28339 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ch' )  ->  E. f ta )
141, 13bnj832 28298 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ta )
15 bnj645 28290 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  et )
16 19.41v 1873 . . . . 5  |-  ( E. f ( ta  /\  et )  <->  ( E. f ta  /\  et ) )
1714, 15, 16sylanbrc 645 . . . 4  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ( ta  /\  et ) )
18 bnj642 28288 . . . 4  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  R  FrSe  A )
19 19.41v 1873 . . . 4  |-  ( E. f ( ( ta 
/\  et )  /\  R  FrSe  A )  <->  ( E. f ( ta  /\  et )  /\  R  FrSe  A ) )
2017, 18, 19sylanbrc 645 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ( ( ta 
/\  et )  /\  R  FrSe  A ) )
21 bnj170 28234 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et )  <->  ( ( ta  /\  et )  /\  R  FrSe  A ) )
2220, 21bnj1198 28339 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ( R  FrSe  A  /\  ta  /\  et ) )
23 bnj908.18 . . . 4  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
24 bnj908.19 . . . 4  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
25 bnj908.1 . . . . . 6  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
2625, 3, 6bnj523 28430 . . . . 5  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
27 bnj908.2 . . . . . 6  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2827, 4, 6bnj539 28434 . . . . 5  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
29 bnj908.3 . . . . 5  |-  D  =  ( om  \  { (/)
} )
30 bnj908.16 . . . . 5  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
3126, 28, 29, 30, 12, 23bnj544 28437 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
3223, 24, 31bnj561 28446 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  G  Fn  n )
33 bnj908.13 . . . . . 6  |-  ( ph"  <->  [. G  / 
f ]. ph )
3430bnj528 28432 . . . . . 6  |-  G  e. 
_V
3525, 33, 34bnj609 28460 . . . . 5  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3626, 29, 30, 12, 23, 31, 35bnj545 28438 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )
3723, 24, 36bnj562 28447 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ph" )
38 bnj908.20 . . . 4  |-  ( ze  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =  suc  i ) )
39 bnj908.22 . . . 4  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
40 bnj908.23 . . . 4  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
41 bnj908.24 . . . 4  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
42 bnj908.25 . . . 4  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
43 bnj908.26 . . . 4  |-  G  =  ( f  u.  { <. m ,  C >. } )
44 bnj908.21 . . . 4  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
45 bnj908.14 . . . . 5  |-  ( ps"  <->  [. G  / 
f ]. ps )
4627, 45, 34bnj611 28461 . . . 4  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
4729, 30, 12, 23, 24, 38, 39, 40, 41, 42, 43, 26, 28, 31, 44, 32, 46bnj571 28449 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  et )  ->  ps" )
4832, 37, 473jca 1132 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  et )  -> 
( G  Fn  n  /\  ph"  /\  ps" ) )
4922, 48bnj593 28285 1  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  ch' 
/\  et )  ->  E. f ( G  Fn  n  /\  ph"  /\  ps" ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1532    = wceq 1633    e. wcel 1701   E!weu 2176    =/= wne 2479   A.wral 2577   [.wsbc 3025    \ cdif 3183    u. cun 3184   (/)c0 3489   {csn 3674   <.cop 3677   U_ciun 3942   class class class wbr 4060    _E cep 4340   suc csuc 4431   omcom 4693    Fn wfn 5287   ` cfv 5292    /\ w-bnj17 28222    predc-bnj14 28224    FrSe w-bnj15 28228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549  ax-reg 7351
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-bnj17 28223  df-bnj14 28225  df-bnj13 28227  df-bnj15 28229
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