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Theorem bnj535 29701
Description: Technical lemma for bnj852 29732. 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
bnj535.1  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj535.2  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj535.3  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj535.4  |-  ( ta  <->  ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m ) )
Assertion
Ref Expression
bnj535  |-  ( ( R  FrSe  A  /\  ta  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m )  ->  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)    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 bnj535
StepHypRef Expression
1 bnj422 29520 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m )  <->  ( n  =  ( m  u. 
{ m } )  /\  f  Fn  m  /\  R  FrSe  A  /\  ta ) )
2 bnj251 29507 . . 3  |-  ( ( n  =  ( m  u.  { m }
)  /\  f  Fn  m  /\  R  FrSe  A  /\  ta )  <->  ( n  =  ( m  u. 
{ m } )  /\  ( f  Fn  m  /\  ( R 
FrSe  A  /\  ta )
) ) )
31, 2bitri 253 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m )  <->  ( n  =  ( m  u. 
{ m } )  /\  ( f  Fn  m  /\  ( R 
FrSe  A  /\  ta )
) ) )
4 fvex 5875 . . . . . . . . 9  |-  ( f `
 p )  e. 
_V
5 bnj535.1 . . . . . . . . . 10  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
6 bnj535.2 . . . . . . . . . 10  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
7 bnj535.4 . . . . . . . . . 10  |-  ( ta  <->  ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m ) )
85, 6, 7bnj518 29697 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta )  ->  A. y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
9 iunexg 6769 . . . . . . . . 9  |-  ( ( ( f `  p
)  e.  _V  /\  A. y  e.  ( f `
 p )  pred ( y ,  A ,  R )  e.  _V )  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V )
104, 8, 9sylancr 669 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta )  ->  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
11 vex 3048 . . . . . . . . 9  |-  m  e. 
_V
1211bnj519 29544 . . . . . . . 8  |-  ( U_ y  e.  ( f `  p )  pred (
y ,  A ,  R )  e.  _V  ->  Fun  { <. m ,  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
>. } )
1310, 12syl 17 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta )  ->  Fun  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
14 dmsnopg 5307 . . . . . . . 8  |-  ( U_ y  e.  ( f `  p )  pred (
y ,  A ,  R )  e.  _V  ->  dom  { <. m ,  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
>. }  =  { m } )
1510, 14syl 17 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta )  ->  dom  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. }  =  { m } )
1613, 15bnj1422 29649 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta )  ->  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. }  Fn  { m } )
17 bnj521 29545 . . . . . . 7  |-  ( m  i^i  { m }
)  =  (/)
18 fnun 5682 . . . . . . 7  |-  ( ( ( f  Fn  m  /\  { <. m ,  U_ y  e.  ( f `  p )  pred (
y ,  A ,  R ) >. }  Fn  { m } )  /\  ( m  i^i  { m } )  =  (/) )  ->  ( f  u. 
{ <. m ,  U_ y  e.  ( f `  p )  pred (
y ,  A ,  R ) >. } )  Fn  ( m  u. 
{ m } ) )
1917, 18mpan2 677 . . . . . 6  |-  ( ( f  Fn  m  /\  {
<. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. }  Fn  { m } )  ->  (
f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )  Fn  (
m  u.  { m } ) )
2016, 19sylan2 477 . . . . 5  |-  ( ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) )  ->  (
f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )  Fn  (
m  u.  { m } ) )
21 bnj535.3 . . . . . 6  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
2221fneq1i 5670 . . . . 5  |-  ( G  Fn  ( m  u. 
{ m } )  <-> 
( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )  Fn  (
m  u.  { m } ) )
2320, 22sylibr 216 . . . 4  |-  ( ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) )  ->  G  Fn  ( m  u.  {
m } ) )
24 fneq2 5665 . . . 4  |-  ( n  =  ( m  u. 
{ m } )  ->  ( G  Fn  n 
<->  G  Fn  ( m  u.  { m }
) ) )
2523, 24syl5ibr 225 . . 3  |-  ( n  =  ( m  u. 
{ m } )  ->  ( ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) )  ->  G  Fn  n ) )
2625imp 431 . 2  |-  ( ( n  =  ( m  u.  { m }
)  /\  ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) ) )  ->  G  Fn  n )
273, 26sylbi 199 1  |-  ( ( R  FrSe  A  /\  ta  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m )  ->  G  Fn  n )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    u. cun 3402    i^i cin 3403   (/)c0 3731   {csn 3968   <.cop 3974   U_ciun 4278   dom cdm 4834   suc csuc 5425   Fun wfun 5576    Fn wfn 5577   ` cfv 5582   omcom 6692    /\ w-bnj17 29491    predc-bnj14 29493    FrSe w-bnj15 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583  ax-reg 8107
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-bnj17 29492  df-bnj14 29494  df-bnj13 29496  df-bnj15 29498
This theorem is referenced by:  bnj543  29704
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