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Theorem bnj910 28491
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
bnj910.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj910.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj910.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj910.4  |-  ( ph'  <->  [. p  /  n ]. ph )
bnj910.5  |-  ( ps'  <->  [. p  /  n ]. ps )
bnj910.6  |-  ( ch'  <->  [. p  /  n ]. ch )
bnj910.7  |-  ( ph"  <->  [. G  / 
f ]. ph' )
bnj910.8  |-  ( ps"  <->  [. G  / 
f ]. ps' )
bnj910.9  |-  ( ch"  <->  [. G  / 
f ]. ch' )
bnj910.10  |-  D  =  ( om  \  { (/)
} )
bnj910.11  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj910.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj910.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj910.14  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj910.15  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
Assertion
Ref Expression
bnj910  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ch" )
Distinct variable groups:    A, f,
i, m, n, y    D, f, i, n    i, G    R, f, i, m, n, y    f, X, i, n    f, p, i, n    ph, i
Allowed substitution hints:    ph( y, f, m, n, p)    ps( y, f, i, m, n, p)    ch( y, f, i, m, n, p)    ta( y, f, i, m, n, p)    si( y, f, i, m, n, p)    A( p)    B( y, f, i, m, n, p)    C( y, f, i, m, n, p)    D( y, m, p)    R( p)    G( y, f, m, n, p)    X( y, m, p)    ph'( y, f, i, m, n, p)    ps'( y, f, i, m, n, p)    ch'( y, f, i, m, n, p)    ph"( y, f, i, m, n, p)   
ps"( y, f, i, m, n, p)    ch"( y, f, i, m, n, p)

Proof of Theorem bnj910
StepHypRef Expression
1 bnj910.3 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj910.10 . . . 4  |-  D  =  ( om  \  { (/)
} )
31, 2bnj970 28490 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  e.  D )
4 bnj910.1 . . . . 5  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
5 bnj910.2 . . . . 5  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
6 bnj910.12 . . . . 5  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
7 bnj910.14 . . . . 5  |-  ( ta  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
8 bnj910.15 . . . . 5  |-  ( si  <->  ( n  e.  D  /\  p  =  suc  n  /\  m  e.  n )
)
94, 5, 1, 2, 6, 7, 8bnj969 28489 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
10 simpr3 963 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  =  suc  n )
111bnj1235 28348 . . . . . 6  |-  ( ch 
->  f  Fn  n
)
12113ad2ant1 976 . . . . 5  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
1312adantl 452 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
f  Fn  n )
14 bnj910.13 . . . . . 6  |-  G  =  ( f  u.  { <. n ,  C >. } )
1514bnj941 28315 . . . . 5  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )
16153impib 1149 . . . 4  |-  ( ( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n )  ->  G  Fn  p )
179, 10, 13, 16syl3anc 1182 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  G  Fn  p )
18 bnj910.4 . . . 4  |-  ( ph'  <->  [. p  /  n ]. ph )
19 bnj910.7 . . . 4  |-  ( ph"  <->  [. G  / 
f ]. ph' )
204, 5, 1, 18, 19, 2, 6, 14, 7, 8bnj944 28481 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ph" )
21 bnj910.5 . . . 4  |-  ( ps'  <->  [. p  /  n ]. ps )
22 bnj910.8 . . . 4  |-  ( ps"  <->  [. G  / 
f ]. ps' )
235, 1, 2, 6, 14, 9bnj967 28488 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
241, 2, 6, 14, 9, 17bnj966 28487 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
255, 1, 21, 22, 6, 14, 23, 24bnj964 28486 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ps" )
263, 17, 20, 25bnj951 28318 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )
27 bnj910.6 . . . 4  |-  ( ch'  <->  [. p  /  n ]. ch )
28 vex 2825 . . . 4  |-  p  e. 
_V
291, 18, 21, 27, 28bnj919 28308 . . 3  |-  ( ch'  <->  (
p  e.  D  /\  f  Fn  p  /\  ph' 
/\  ps' ) )
30 bnj910.9 . . 3  |-  ( ch"  <->  [. G  / 
f ]. ch' )
3114bnj918 28307 . . 3  |-  G  e. 
_V
3229, 19, 22, 30, 31bnj976 28320 . 2  |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )
3326, 32sylibr 203 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ch" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   {cab 2302   A.wral 2577   E.wrex 2578   _Vcvv 2822   [.wsbc 3025    \ cdif 3183    u. cun 3184   (/)c0 3489   {csn 3674   <.cop 3677   U_ciun 3942   suc csuc 4431   omcom 4693    Fn wfn 5287   ` cfv 5292    /\ w-bnj17 28222    predc-bnj14 28224    FrSe w-bnj15 28228
This theorem is referenced by:  bnj998  28499
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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