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Theorem bnj535 28922
Description: Technical lemma for bnj852 28953. 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 28740 . . 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 28727 . . 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 240 . 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 5539 . . . . . . . . 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 28918 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta )  ->  A. y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
9 iunexg 5767 . . . . . . . . 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 644 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta )  ->  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
11 vex 2791 . . . . . . . . 9  |-  m  e. 
_V
1211bnj519 28764 . . . . . . . 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 15 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta )  ->  Fun  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
14 dmsnopg 5144 . . . . . . . 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 15 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta )  ->  dom  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. }  =  { m } )
1613, 15bnj1422 28870 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta )  ->  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. }  Fn  { m } )
17 bnj521 28765 . . . . . . 7  |-  ( m  i^i  { m }
)  =  (/)
18 fnun 5350 . . . . . . 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 652 . . . . . 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 460 . . . . 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 5338 . . . . 5  |-  ( G  Fn  ( m  u. 
{ m } )  <-> 
( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )  Fn  (
m  u.  { m } ) )
2320, 22sylibr 203 . . . 4  |-  ( ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) )  ->  G  Fn  ( m  u.  {
m } ) )
24 fneq2 5334 . . . 4  |-  ( n  =  ( m  u. 
{ m } )  ->  ( G  Fn  n 
<->  G  Fn  ( m  u.  { m }
) ) )
2523, 24syl5ibr 212 . . 3  |-  ( n  =  ( m  u. 
{ m } )  ->  ( ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) )  ->  G  Fn  n ) )
2625imp 418 . 2  |-  ( ( n  =  ( m  u.  { m }
)  /\  ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) ) )  ->  G  Fn  n )
273, 26sylbi 187 1  |-  ( ( R  FrSe  A  /\  ta  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m )  ->  G  Fn  n )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   <.cop 3643   U_ciun 3905   suc csuc 4394   omcom 4656   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj543  28925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718
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