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Theorem bnj543 29241
Description: Technical lemma for bnj852 29269. 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 29048 . . . . . . 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 29050 . . . . . . 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 240 . . . . . 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 29045 . . . . . 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 29047 . . . . . 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 266 . . . . 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 29047 . . . . . 6  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ( ph' 
/\  ps' )  /\  (
m  e.  om  /\  p  e.  m )
) )
873anbi1i 1142 . . . . 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 29039 . . . . . . 7  |-  ( ( f  Fn  m  /\  ph' 
/\  ps' )  <->  ( ( ph' 
/\  ps' )  /\  f  Fn  m ) )
119, 10bitri 240 . . . . . 6  |-  ( ta  <->  ( ( ph'  /\  ps' )  /\  f  Fn  m )
)
12 bnj543.5 . . . . . . 7  |-  ( si  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )
)
13 3anan32 946 . . . . . . 7  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )  <->  ( ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
1412, 13bitri 240 . . . . . 6  |-  ( si  <->  ( ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
1511, 14anbi12i 678 . . . . 5  |-  ( ( ta  /\  si )  <->  ( ( ( ph'  /\  ps' )  /\  f  Fn  m )  /\  ( ( m  e. 
om  /\  p  e.  m )  /\  n  =  suc  m ) ) )
166, 8, 153bitr4ri 269 . . . 4  |-  ( ( ta  /\  si )  <->  ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m ) )
1716anbi2i 675 . . 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 938 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  <->  ( R  FrSe  A  /\  ( ta 
/\  si ) ) )
19 bnj252 29044 . . 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 268 . 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 4414 . . . . . . 7  |-  suc  m  =  ( m  u. 
{ m } )
2221eqeq2i 2306 . . . . . 6  |-  ( n  =  suc  m  <->  n  =  ( m  u.  { m } ) )
23223anbi2i 1143 . . . . 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 675 . . . 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 29044 . . . 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 268 . . 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 227 . . . 4  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )
3127, 28, 29, 30bnj535 29238 . . 3  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  ( m  u. 
{ m } )  /\  f  Fn  m
)  ->  G  Fn  n )
3226, 31sylbi 187 . 2  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m )  ->  G  Fn  n )
3320, 32sylbi 187 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    u. cun 3163   (/)c0 3468   {csn 3653   <.cop 3656   U_ciun 3921   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj544  29242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-reg 7322
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034
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