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Theorem bnj543 34052
Description: Technical lemma for bnj852 34080. 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
bnj543.1  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj543.2  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj543.3  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj543.4  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj543.5  |-  ( si  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )
)
Assertion
Ref Expression
bnj543  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Distinct variable groups:    A, i, p, y    R, i, p, y    f, i, p, y    i, m, p   
p, ph'
Allowed substitution hints:    ta( x, y, f, i, m, n, p)    si( x, y, f, i, m, n, p)    A( x, f, m, n)    R( x, f, m, n)    G( x, y, f, i, m, n, p)    ph'( x, y, f, i, m, n)    ps'( x, y, f, i, m, n, p)

Proof of Theorem bnj543
StepHypRef Expression
1 bnj257 33860 . . . . . . 7  |-  ( ( ( ph'  /\  ps' )  /\  ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m
)  <->  ( ( ph'  /\  ps' )  /\  (
m  e.  om  /\  p  e.  m )  /\  f  Fn  m  /\  n  =  suc  m ) )
2 bnj268 33862 . . . . . . 7  |-  ( ( ( ph'  /\  ps' )  /\  ( m  e.  om  /\  p  e.  m )  /\  f  Fn  m  /\  n  =  suc  m )  <->  ( ( ph' 
/\  ps' )  /\  f  Fn  m  /\  (
m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
31, 2bitri 249 . . . . . 6  |-  ( ( ( ph'  /\  ps' )  /\  ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m
)  <->  ( ( ph'  /\  ps' )  /\  f  Fn  m  /\  (
m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
4 bnj253 33857 . . . . . 6  |-  ( ( ( ph'  /\  ps' )  /\  ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m
)  <->  ( ( ( ph'  /\  ps' )  /\  (
m  e.  om  /\  p  e.  m )
)  /\  n  =  suc  m  /\  f  Fn  m ) )
5 bnj256 33859 . . . . . 6  |-  ( ( ( ph'  /\  ps' )  /\  f  Fn  m  /\  ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m )  <->  ( (
( ph'  /\  ps' )  /\  f  Fn  m )  /\  ( ( m  e. 
om  /\  p  e.  m )  /\  n  =  suc  m ) ) )
63, 4, 53bitr3i 275 . . . . 5  |-  ( ( ( ( ph'  /\  ps' )  /\  ( m  e.  om  /\  p  e.  m ) )  /\  n  =  suc  m  /\  f  Fn  m )  <->  ( (
( ph'  /\  ps' )  /\  f  Fn  m )  /\  ( ( m  e. 
om  /\  p  e.  m )  /\  n  =  suc  m ) ) )
7 bnj256 33859 . . . . . 6  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ( ph' 
/\  ps' )  /\  (
m  e.  om  /\  p  e.  m )
) )
873anbi1i 1187 . . . . 5  |-  ( ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m )  <->  ( (
( ph'  /\  ps' )  /\  ( m  e.  om  /\  p  e.  m ) )  /\  n  =  suc  m  /\  f  Fn  m ) )
9 bnj543.4 . . . . . . 7  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
10 bnj170 33851 . . . . . . 7  |-  ( ( f  Fn  m  /\  ph' 
/\  ps' )  <->  ( ( ph' 
/\  ps' )  /\  f  Fn  m ) )
119, 10bitri 249 . . . . . 6  |-  ( ta  <->  ( ( ph'  /\  ps' )  /\  f  Fn  m )
)
12 bnj543.5 . . . . . . 7  |-  ( si  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )
)
13 3anan32 985 . . . . . . 7  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )  <->  ( ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
1412, 13bitri 249 . . . . . 6  |-  ( si  <->  ( ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
1511, 14anbi12i 697 . . . . 5  |-  ( ( ta  /\  si )  <->  ( ( ( ph'  /\  ps' )  /\  f  Fn  m )  /\  ( ( m  e. 
om  /\  p  e.  m )  /\  n  =  suc  m ) ) )
166, 8, 153bitr4ri 278 . . . 4  |-  ( ( ta  /\  si )  <->  ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m ) )
1716anbi2i 694 . . 3  |-  ( ( R  FrSe  A  /\  ( ta  /\  si )
)  <->  ( R  FrSe  A  /\  ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m
) ) )
18 3anass 977 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  <->  ( R  FrSe  A  /\  ( ta 
/\  si ) ) )
19 bnj252 33856 . . 3  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m )  <->  ( R  FrSe  A  /\  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m ) ) )
2017, 18, 193bitr4i 277 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  <->  ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m
) )
21 df-suc 4893 . . . . . . 7  |-  suc  m  =  ( m  u. 
{ m } )
2221eqeq2i 2475 . . . . . 6  |-  ( n  =  suc  m  <->  n  =  ( m  u.  { m } ) )
23223anbi2i 1188 . . . . 5  |-  ( ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m )  <->  ( ( ph' 
/\  ps'  /\  m  e. 
om  /\  p  e.  m )  /\  n  =  ( m  u. 
{ m } )  /\  f  Fn  m
) )
2423anbi2i 694 . . . 4  |-  ( ( R  FrSe  A  /\  ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m ) )  <->  ( R  FrSe  A  /\  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  /\  n  =  ( m  u. 
{ m } )  /\  f  Fn  m
) ) )
25 bnj252 33856 . . . 4  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  ( m  u. 
{ m } )  /\  f  Fn  m
)  <->  ( R  FrSe  A  /\  ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m ) ) )
2624, 19, 253bitr4i 277 . . 3  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m )  <->  ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m ) )
27 bnj543.1 . . . 4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
28 bnj543.2 . . . 4  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
29 bnj543.3 . . . 4  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
30 biid 236 . . . 4  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )
3127, 28, 29, 30bnj535 34049 . . 3  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  ( m  u. 
{ m } )  /\  f  Fn  m
)  ->  G  Fn  n )
3226, 31sylbi 195 . 2  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m )  ->  G  Fn  n )
3320, 32sylbi 195 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    u. cun 3469   (/)c0 3793   {csn 4032   <.cop 4038   U_ciun 4332   suc csuc 4889    Fn wfn 5589   ` cfv 5594   omcom 6699    /\ w-bnj17 33839    predc-bnj14 33841    FrSe w-bnj15 33845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591  ax-reg 8036
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-bnj17 33840  df-bnj14 33842  df-bnj13 33844  df-bnj15 33846
This theorem is referenced by:  bnj544  34053
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