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Theorem bnj543 29699
Description: Technical lemma for bnj852 29727. 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 29507 . . . . . . 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 29509 . . . . . . 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 252 . . . . . 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 29504 . . . . . 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 29506 . . . . . 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 278 . . . . 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 29506 . . . . . 6  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ( ph' 
/\  ps' )  /\  (
m  e.  om  /\  p  e.  m )
) )
873anbi1i 1196 . . . . 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 29498 . . . . . . 7  |-  ( ( f  Fn  m  /\  ph' 
/\  ps' )  <->  ( ( ph' 
/\  ps' )  /\  f  Fn  m ) )
119, 10bitri 252 . . . . . 6  |-  ( ta  <->  ( ( ph'  /\  ps' )  /\  f  Fn  m )
)
12 bnj543.5 . . . . . . 7  |-  ( si  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )
)
13 3anan32 994 . . . . . . 7  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )  <->  ( ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
1412, 13bitri 252 . . . . . 6  |-  ( si  <->  ( ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
1511, 14anbi12i 701 . . . . 5  |-  ( ( ta  /\  si )  <->  ( ( ( ph'  /\  ps' )  /\  f  Fn  m )  /\  ( ( m  e. 
om  /\  p  e.  m )  /\  n  =  suc  m ) ) )
166, 8, 153bitr4ri 281 . . . 4  |-  ( ( ta  /\  si )  <->  ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m ) )
1716anbi2i 698 . . 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 986 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  <->  ( R  FrSe  A  /\  ( ta 
/\  si ) ) )
19 bnj252 29503 . . 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 280 . 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 5444 . . . . . . 7  |-  suc  m  =  ( m  u. 
{ m } )
2221eqeq2i 2440 . . . . . 6  |-  ( n  =  suc  m  <->  n  =  ( m  u.  { m } ) )
23223anbi2i 1197 . . . . 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 698 . . . 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 29503 . . . 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 280 . . 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 239 . . . 4  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )
3127, 28, 29, 30bnj535 29696 . . 3  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  ( m  u. 
{ m } )  /\  f  Fn  m
)  ->  G  Fn  n )
3226, 31sylbi 198 . 2  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m )  ->  G  Fn  n )
3320, 32sylbi 198 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775    u. cun 3434   (/)c0 3761   {csn 3996   <.cop 4002   U_ciun 4296   suc csuc 5440    Fn wfn 5592   ` cfv 5597   omcom 6702    /\ w-bnj17 29486    predc-bnj14 29488    FrSe w-bnj15 29492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-un 6593  ax-reg 8109
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-om 6703  df-bnj17 29487  df-bnj14 29489  df-bnj13 29491  df-bnj15 29493
This theorem is referenced by:  bnj544  29700
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