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Theorem tfrlemi1 6001
Description: We can define an acceptable function on any ordinal.

As with many of the transfinite recursion theorems, we have a hypothesis that states that  F is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses
Ref Expression
tfrlemisucfn.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlemisucfn.2  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
Assertion
Ref Expression
tfrlemi1  |-  ( (
ph  /\  C  e.  On )  ->  E. g
( g  Fn  C  /\  A. u  e.  C  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )
Distinct variable groups:    f, g, u, x, y, A    f, F, g, u, x, y    ph, y    C, g, u    ph, f
Allowed substitution hints:    ph( x, u, g)    C( x, y, f)

Proof of Theorem tfrlemi1
Dummy variables  e  h  k  t  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 108 . . . . . . 7  |-  ( ( z  =  w  /\  g  =  k )  ->  g  =  k )
2 simpl 107 . . . . . . 7  |-  ( ( z  =  w  /\  g  =  k )  ->  z  =  w )
31, 2fneq12d 5042 . . . . . 6  |-  ( ( z  =  w  /\  g  =  k )  ->  ( g  Fn  z  <->  k  Fn  w ) )
41fveq1d 5231 . . . . . . . 8  |-  ( ( z  =  w  /\  g  =  k )  ->  ( g `  u
)  =  ( k `
 u ) )
51reseq1d 4659 . . . . . . . . 9  |-  ( ( z  =  w  /\  g  =  k )  ->  ( g  |`  u
)  =  ( k  |`  u ) )
65fveq2d 5233 . . . . . . . 8  |-  ( ( z  =  w  /\  g  =  k )  ->  ( F `  (
g  |`  u ) )  =  ( F `  ( k  |`  u
) ) )
74, 6eqeq12d 2097 . . . . . . 7  |-  ( ( z  =  w  /\  g  =  k )  ->  ( ( g `  u )  =  ( F `  ( g  |`  u ) )  <->  ( k `  u )  =  ( F `  ( k  |`  u ) ) ) )
82, 7raleqbidv 2566 . . . . . 6  |-  ( ( z  =  w  /\  g  =  k )  ->  ( A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u ) )  <->  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) ) )
93, 8anbi12d 457 . . . . 5  |-  ( ( z  =  w  /\  g  =  k )  ->  ( ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u ) ) )  <-> 
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) ) )
109cbvexdva 1847 . . . 4  |-  ( z  =  w  ->  ( E. g ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u ) ) )  <->  E. k ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) ) ) )
1110imbi2d 228 . . 3  |-  ( z  =  w  ->  (
( ph  ->  E. g
( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )  <->  ( ph  ->  E. k ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) ) ) ) )
12 fneq2 5039 . . . . . 6  |-  ( z  =  C  ->  (
g  Fn  z  <->  g  Fn  C ) )
13 raleq 2554 . . . . . 6  |-  ( z  =  C  ->  ( A. u  e.  z 
( g `  u
)  =  ( F `
 ( g  |`  u ) )  <->  A. u  e.  C  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1412, 13anbi12d 457 . . . . 5  |-  ( z  =  C  ->  (
( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) )  <-> 
( g  Fn  C  /\  A. u  e.  C  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) ) )
1514exbidv 1748 . . . 4  |-  ( z  =  C  ->  ( E. g ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u ) ) )  <->  E. g ( g  Fn  C  /\  A. u  e.  C  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) ) )
1615imbi2d 228 . . 3  |-  ( z  =  C  ->  (
( ph  ->  E. g
( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )  <->  ( ph  ->  E. g ( g  Fn  C  /\  A. u  e.  C  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) ) ) )
17 r19.21v 2443 . . . 4  |-  ( A. w  e.  z  ( ph  ->  E. k ( k  Fn  w  /\  A. u  e.  w  (
k `  u )  =  ( F `  ( k  |`  u
) ) ) )  <-> 
( ph  ->  A. w  e.  z  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) ) )
18 tfrlemisucfn.1 . . . . . . . . 9  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
1918tfrlem3 5980 . . . . . . . 8  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. e  e.  z  ( g `  e )  =  ( F `  ( g  |`  e
) ) ) }
20 tfrlemisucfn.2 . . . . . . . . . 10  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
21 fveq2 5229 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
2221eleq1d 2151 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( F `  x
)  e.  _V  <->  ( F `  z )  e.  _V ) )
2322anbi2d 452 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( Fun  F  /\  ( F `  x )  e.  _V )  <->  ( Fun  F  /\  ( F `  z )  e.  _V ) ) )
2423cbvalv 1837 . . . . . . . . . 10  |-  ( A. x ( Fun  F  /\  ( F `  x
)  e.  _V )  <->  A. z ( Fun  F  /\  ( F `  z
)  e.  _V )
)
2520, 24sylib 120 . . . . . . . . 9  |-  ( ph  ->  A. z ( Fun 
F  /\  ( F `  z )  e.  _V ) )
2625adantr 270 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  On  /\  A. w  e.  z  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) ) )  ->  A. z
( Fun  F  /\  ( F `  z )  e.  _V ) )
27 simpr 108 . . . . . . . . . . . . 13  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  k  =  f )
28 simplr 497 . . . . . . . . . . . . 13  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  w  =  v )
2927, 28fneq12d 5042 . . . . . . . . . . . 12  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
k  Fn  w  <->  f  Fn  v ) )
3027eleq1d 2151 . . . . . . . . . . . 12  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
k  e.  A  <->  f  e.  A ) )
31 simpll 496 . . . . . . . . . . . . 13  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  t  =  h )
3227fveq2d 5233 . . . . . . . . . . . . . . . 16  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  ( F `  k )  =  ( F `  f ) )
3328, 32opeq12d 3598 . . . . . . . . . . . . . . 15  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  <. w ,  ( F `  k ) >.  =  <. v ,  ( F `  f ) >. )
3433sneqd 3429 . . . . . . . . . . . . . 14  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  { <. w ,  ( F `  k ) >. }  =  { <. v ,  ( F `  f )
>. } )
3527, 34uneq12d 3137 . . . . . . . . . . . . 13  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
k  u.  { <. w ,  ( F `  k ) >. } )  =  ( f  u. 
{ <. v ,  ( F `  f )
>. } ) )
3631, 35eqeq12d 2097 . . . . . . . . . . . 12  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
t  =  ( k  u.  { <. w ,  ( F `  k ) >. } )  <-> 
h  =  ( f  u.  { <. v ,  ( F `  f ) >. } ) ) )
3729, 30, 363anbi123d 1244 . . . . . . . . . . 11  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
( k  Fn  w  /\  k  e.  A  /\  t  =  (
k  u.  { <. w ,  ( F `  k ) >. } ) )  <->  ( f  Fn  v  /\  f  e.  A  /\  h  =  ( f  u.  { <. v ,  ( F `
 f ) >. } ) ) ) )
3837cbvexdva 1847 . . . . . . . . . 10  |-  ( ( t  =  h  /\  w  =  v )  ->  ( E. k ( k  Fn  w  /\  k  e.  A  /\  t  =  ( k  u.  { <. w ,  ( F `  k )
>. } ) )  <->  E. f
( f  Fn  v  /\  f  e.  A  /\  h  =  (
f  u.  { <. v ,  ( F `  f ) >. } ) ) ) )
3938cbvrexdva 2589 . . . . . . . . 9  |-  ( t  =  h  ->  ( E. w  e.  z  E. k ( k  Fn  w  /\  k  e.  A  /\  t  =  ( k  u.  { <. w ,  ( F `
 k ) >. } ) )  <->  E. v  e.  z  E. f
( f  Fn  v  /\  f  e.  A  /\  h  =  (
f  u.  { <. v ,  ( F `  f ) >. } ) ) ) )
4039cbvabv 2206 . . . . . . . 8  |-  { t  |  E. w  e.  z  E. k ( k  Fn  w  /\  k  e.  A  /\  t  =  ( k  u.  { <. w ,  ( F `  k )
>. } ) ) }  =  { h  |  E. v  e.  z  E. f ( f  Fn  v  /\  f  e.  A  /\  h  =  ( f  u. 
{ <. v ,  ( F `  f )
>. } ) ) }
41 simpl 107 . . . . . . . . 9  |-  ( ( z  e.  On  /\  A. w  e.  z  E. k ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) ) )  ->  z  e.  On )
4241adantl 271 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  On  /\  A. w  e.  z  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) ) )  ->  z  e.  On )
43 simpr 108 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  A. w  e.  z  E. k ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) ) )  ->  A. w  e.  z  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) )
44 simpr 108 . . . . . . . . . . . . . 14  |-  ( ( w  =  v  /\  k  =  f )  ->  k  =  f )
45 simpl 107 . . . . . . . . . . . . . 14  |-  ( ( w  =  v  /\  k  =  f )  ->  w  =  v )
4644, 45fneq12d 5042 . . . . . . . . . . . . 13  |-  ( ( w  =  v  /\  k  =  f )  ->  ( k  Fn  w  <->  f  Fn  v ) )
47 simplr 497 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  k  =  f )
48 simpr 108 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  u  =  y )
4947, 48fveq12d 5235 . . . . . . . . . . . . . . 15  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  (
k `  u )  =  ( f `  y ) )
5047, 48reseq12d 4661 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  (
k  |`  u )  =  ( f  |`  y
) )
5150fveq2d 5233 . . . . . . . . . . . . . . 15  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  ( F `  ( k  |`  u ) )  =  ( F `  (
f  |`  y ) ) )
5249, 51eqeq12d 2097 . . . . . . . . . . . . . 14  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  (
( k `  u
)  =  ( F `
 ( k  |`  u ) )  <->  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
53 simpll 496 . . . . . . . . . . . . . 14  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  w  =  v )
5452, 53cbvraldva2 2586 . . . . . . . . . . . . 13  |-  ( ( w  =  v  /\  k  =  f )  ->  ( A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) )  <->  A. y  e.  v  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
5546, 54anbi12d 457 . . . . . . . . . . . 12  |-  ( ( w  =  v  /\  k  =  f )  ->  ( ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) )  <-> 
( f  Fn  v  /\  A. y  e.  v  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) ) )
5655cbvexdva 1847 . . . . . . . . . . 11  |-  ( w  =  v  ->  ( E. k ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) )  <->  E. f ( f  Fn  v  /\  A. y  e.  v  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) ) )
5756cbvralv 2582 . . . . . . . . . 10  |-  ( A. w  e.  z  E. k ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) )  <->  A. v  e.  z  E. f ( f  Fn  v  /\  A. y  e.  v  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
5843, 57sylib 120 . . . . . . . . 9  |-  ( ( z  e.  On  /\  A. w  e.  z  E. k ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) ) )  ->  A. v  e.  z  E. f
( f  Fn  v  /\  A. y  e.  v  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )
5958adantl 271 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  On  /\  A. w  e.  z  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) ) )  ->  A. v  e.  z  E. f
( f  Fn  v  /\  A. y  e.  v  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )
6019, 26, 40, 42, 59tfrlemiex 6000 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  On  /\  A. w  e.  z  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) ) )  ->  E. g
( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )
6160expr 367 . . . . . 6  |-  ( (
ph  /\  z  e.  On )  ->  ( A. w  e.  z  E. k ( k  Fn  w  /\  A. u  e.  w  ( k `  u )  =  ( F `  ( k  |`  u ) ) )  ->  E. g ( g  Fn  z  /\  A. u  e.  z  (
g `  u )  =  ( F `  ( g  |`  u
) ) ) ) )
6261expcom 114 . . . . 5  |-  ( z  e.  On  ->  ( ph  ->  ( A. w  e.  z  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) )  ->  E. g ( g  Fn  z  /\  A. u  e.  z  (
g `  u )  =  ( F `  ( g  |`  u
) ) ) ) ) )
6362a2d 26 . . . 4  |-  ( z  e.  On  ->  (
( ph  ->  A. w  e.  z  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) )  ->  ( ph  ->  E. g ( g  Fn  z  /\  A. u  e.  z  (
g `  u )  =  ( F `  ( g  |`  u
) ) ) ) ) )
6417, 63syl5bi 150 . . 3  |-  ( z  e.  On  ->  ( A. w  e.  z 
( ph  ->  E. k
( k  Fn  w  /\  A. u  e.  w  ( k `  u
)  =  ( F `
 ( k  |`  u ) ) ) )  ->  ( ph  ->  E. g ( g  Fn  z  /\  A. u  e.  z  (
g `  u )  =  ( F `  ( g  |`  u
) ) ) ) ) )
6511, 16, 64tfis3 4355 . 2  |-  ( C  e.  On  ->  ( ph  ->  E. g ( g  Fn  C  /\  A. u  e.  C  (
g `  u )  =  ( F `  ( g  |`  u
) ) ) ) )
6665impcom 123 1  |-  ( (
ph  /\  C  e.  On )  ->  E. g
( g  Fn  C  /\  A. u  e.  C  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2610    u. cun 2980   {csn 3416   <.cop 3419   Oncon0 4146    |` cres 4393   Fun wfun 4946    Fn wfn 4947   ` cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-recs 5974
This theorem is referenced by:  tfrlemi14d  6002  tfrexlem  6003
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