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Theorem bnj535 29263
Description: Technical lemma for bnj852 29294. 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 29081 . . 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 29068 . . 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 242 . 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 5744 . . . . . . . . 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 29259 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta )  ->  A. y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
9 iunexg 5989 . . . . . . . . 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 646 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta )  ->  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
11 vex 2961 . . . . . . . . 9  |-  m  e. 
_V
1211bnj519 29105 . . . . . . . 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 16 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta )  ->  Fun  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
14 dmsnopg 5343 . . . . . . . 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 16 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta )  ->  dom  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. }  =  { m } )
1613, 15bnj1422 29211 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta )  ->  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. }  Fn  { m } )
17 bnj521 29106 . . . . . . 7  |-  ( m  i^i  { m }
)  =  (/)
18 fnun 5553 . . . . . . 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 654 . . . . . 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 462 . . . . 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 5541 . . . . 5  |-  ( G  Fn  ( m  u. 
{ m } )  <-> 
( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )  Fn  (
m  u.  { m } ) )
2320, 22sylibr 205 . . . 4  |-  ( ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) )  ->  G  Fn  ( m  u.  {
m } ) )
24 fneq2 5537 . . . 4  |-  ( n  =  ( m  u. 
{ m } )  ->  ( G  Fn  n 
<->  G  Fn  ( m  u.  { m }
) ) )
2523, 24syl5ibr 214 . . 3  |-  ( n  =  ( m  u. 
{ m } )  ->  ( ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) )  ->  G  Fn  n ) )
2625imp 420 . 2  |-  ( ( n  =  ( m  u.  { m }
)  /\  ( f  Fn  m  /\  ( R  FrSe  A  /\  ta ) ) )  ->  G  Fn  n )
273, 26sylbi 189 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    u. cun 3320    i^i cin 3321   (/)c0 3630   {csn 3816   <.cop 3819   U_ciun 4095   suc csuc 4585   omcom 4847   dom cdm 4880   Fun wfun 5450    Fn wfn 5451   ` cfv 5456    /\ w-bnj17 29052    predc-bnj14 29054    FrSe w-bnj15 29058
This theorem is referenced by:  bnj543  29266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703  ax-reg 7562
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-bnj17 29053  df-bnj14 29055  df-bnj13 29057  df-bnj15 29059
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