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Theorem bnj543 28604
Description: Technical lemma for bnj852 28632. 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 28411 . . . . . . 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 28413 . . . . . . 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 241 . . . . . 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 28408 . . . . . 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 28410 . . . . . 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 267 . . . . 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 28410 . . . . . 6  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ( ph' 
/\  ps' )  /\  (
m  e.  om  /\  p  e.  m )
) )
873anbi1i 1144 . . . . 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 28402 . . . . . . 7  |-  ( ( f  Fn  m  /\  ph' 
/\  ps' )  <->  ( ( ph' 
/\  ps' )  /\  f  Fn  m ) )
119, 10bitri 241 . . . . . 6  |-  ( ta  <->  ( ( ph'  /\  ps' )  /\  f  Fn  m )
)
12 bnj543.5 . . . . . . 7  |-  ( si  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )
)
13 3anan32 948 . . . . . . 7  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )  <->  ( ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
1412, 13bitri 241 . . . . . 6  |-  ( si  <->  ( ( m  e.  om  /\  p  e.  m )  /\  n  =  suc  m ) )
1511, 14anbi12i 679 . . . . 5  |-  ( ( ta  /\  si )  <->  ( ( ( ph'  /\  ps' )  /\  f  Fn  m )  /\  ( ( m  e. 
om  /\  p  e.  m )  /\  n  =  suc  m ) ) )
166, 8, 153bitr4ri 270 . . . 4  |-  ( ( ta  /\  si )  <->  ( ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m ) )
1716anbi2i 676 . . 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 940 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  <->  ( R  FrSe  A  /\  ( ta 
/\  si ) ) )
19 bnj252 28407 . . 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 269 . 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 4530 . . . . . . 7  |-  suc  m  =  ( m  u. 
{ m } )
2221eqeq2i 2399 . . . . . 6  |-  ( n  =  suc  m  <->  n  =  ( m  u.  { m } ) )
23223anbi2i 1145 . . . . 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 676 . . . 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 28407 . . . 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 269 . . 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 228 . . . 4  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )
3127, 28, 29, 30bnj535 28601 . . 3  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  ( m  u. 
{ m } )  /\  f  Fn  m
)  ->  G  Fn  n )
3226, 31sylbi 188 . 2  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m )  /\  n  =  suc  m  /\  f  Fn  m )  ->  G  Fn  n )
3320, 32sylbi 188 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651    u. cun 3263   (/)c0 3573   {csn 3759   <.cop 3762   U_ciun 4037   suc csuc 4526   omcom 4787    Fn wfn 5391   ` cfv 5396    /\ w-bnj17 28390    predc-bnj14 28392    FrSe w-bnj15 28396
This theorem is referenced by:  bnj544  28605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643  ax-reg 7495
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-bnj17 28391  df-bnj14 28393  df-bnj13 28395  df-bnj15 28397
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