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Theorem tfrlemi1 5974
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 107 . . . . . . 7  |-  ( ( z  =  w  /\  g  =  k )  ->  g  =  k )
2 simpl 106 . . . . . . 7  |-  ( ( z  =  w  /\  g  =  k )  ->  z  =  w )
31, 2fneq12d 5016 . . . . . 6  |-  ( ( z  =  w  /\  g  =  k )  ->  ( g  Fn  z  <->  k  Fn  w ) )
41fveq1d 5205 . . . . . . . 8  |-  ( ( z  =  w  /\  g  =  k )  ->  ( g `  u
)  =  ( k `
 u ) )
51reseq1d 4636 . . . . . . . . 9  |-  ( ( z  =  w  /\  g  =  k )  ->  ( g  |`  u
)  =  ( k  |`  u ) )
65fveq2d 5207 . . . . . . . 8  |-  ( ( z  =  w  /\  g  =  k )  ->  ( F `  (
g  |`  u ) )  =  ( F `  ( k  |`  u
) ) )
74, 6eqeq12d 2068 . . . . . . 7  |-  ( ( z  =  w  /\  g  =  k )  ->  ( ( g `  u )  =  ( F `  ( g  |`  u ) )  <->  ( k `  u )  =  ( F `  ( k  |`  u ) ) ) )
82, 7raleqbidv 2532 . . . . . 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 450 . . . . 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 1818 . . . 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 223 . . 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 5013 . . . . . 6  |-  ( z  =  C  ->  (
g  Fn  z  <->  g  Fn  C ) )
13 raleq 2520 . . . . . 6  |-  ( z  =  C  ->  ( A. u  e.  z 
( g `  u
)  =  ( F `
 ( g  |`  u ) )  <->  A. u  e.  C  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1412, 13anbi12d 450 . . . . 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 1720 . . . 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 223 . . 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 2411 . . . 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 5954 . . . . . . . 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 5203 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
2221eleq1d 2120 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( F `  x
)  e.  _V  <->  ( F `  z )  e.  _V ) )
2322anbi2d 445 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( Fun  F  /\  ( F `  x )  e.  _V )  <->  ( Fun  F  /\  ( F `  z )  e.  _V ) ) )
2423cbvalv 1808 . . . . . . . . . 10  |-  ( A. x ( Fun  F  /\  ( F `  x
)  e.  _V )  <->  A. z ( Fun  F  /\  ( F `  z
)  e.  _V )
)
2520, 24sylib 131 . . . . . . . . 9  |-  ( ph  ->  A. z ( Fun 
F  /\  ( F `  z )  e.  _V ) )
2625adantr 265 . . . . . . . 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 107 . . . . . . . . . . . . 13  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  k  =  f )
28 simplr 490 . . . . . . . . . . . . 13  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  w  =  v )
2927, 28fneq12d 5016 . . . . . . . . . . . 12  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
k  Fn  w  <->  f  Fn  v ) )
3027eleq1d 2120 . . . . . . . . . . . 12  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
k  e.  A  <->  f  e.  A ) )
31 simpll 489 . . . . . . . . . . . . 13  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  t  =  h )
3227fveq2d 5207 . . . . . . . . . . . . . . . 16  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  ( F `  k )  =  ( F `  f ) )
3328, 32opeq12d 3582 . . . . . . . . . . . . . . 15  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  <. w ,  ( F `  k ) >.  =  <. v ,  ( F `  f ) >. )
3433sneqd 3413 . . . . . . . . . . . . . 14  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  { <. w ,  ( F `  k ) >. }  =  { <. v ,  ( F `  f )
>. } )
3527, 34uneq12d 3123 . . . . . . . . . . . . 13  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
k  u.  { <. w ,  ( F `  k ) >. } )  =  ( f  u. 
{ <. v ,  ( F `  f )
>. } ) )
3631, 35eqeq12d 2068 . . . . . . . . . . . 12  |-  ( ( ( t  =  h  /\  w  =  v )  /\  k  =  f )  ->  (
t  =  ( k  u.  { <. w ,  ( F `  k ) >. } )  <-> 
h  =  ( f  u.  { <. v ,  ( F `  f ) >. } ) ) )
3729, 30, 363anbi123d 1216 . . . . . . . . . . 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 1818 . . . . . . . . . 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 2555 . . . . . . . . 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 2175 . . . . . . . 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 106 . . . . . . . . 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 266 . . . . . . . 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 107 . . . . . . . . . 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 107 . . . . . . . . . . . . . 14  |-  ( ( w  =  v  /\  k  =  f )  ->  k  =  f )
45 simpl 106 . . . . . . . . . . . . . 14  |-  ( ( w  =  v  /\  k  =  f )  ->  w  =  v )
4644, 45fneq12d 5016 . . . . . . . . . . . . 13  |-  ( ( w  =  v  /\  k  =  f )  ->  ( k  Fn  w  <->  f  Fn  v ) )
47 simplr 490 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  k  =  f )
48 simpr 107 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  u  =  y )
4947, 48fveq12d 5209 . . . . . . . . . . . . . . 15  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  (
k `  u )  =  ( f `  y ) )
5047, 48reseq12d 4638 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  (
k  |`  u )  =  ( f  |`  y
) )
5150fveq2d 5207 . . . . . . . . . . . . . . 15  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  ( F `  ( k  |`  u ) )  =  ( F `  (
f  |`  y ) ) )
5249, 51eqeq12d 2068 . . . . . . . . . . . . . 14  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  (
( k `  u
)  =  ( F `
 ( k  |`  u ) )  <->  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
53 simpll 489 . . . . . . . . . . . . . 14  |-  ( ( ( w  =  v  /\  k  =  f )  /\  u  =  y )  ->  w  =  v )
5452, 53cbvraldva2 2552 . . . . . . . . . . . . 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 450 . . . . . . . . . . . 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 1818 . . . . . . . . . . 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 2548 . . . . . . . . . 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 131 . . . . . . . . 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 266 . . . . . . . 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 5973 . . . . . . 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 361 . . . . . 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 113 . . . . 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 145 . . 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 4334 . 2  |-  ( C  e.  On  ->  ( ph  ->  E. g ( g  Fn  C  /\  A. u  e.  C  (
g `  u )  =  ( F `  ( g  |`  u
) ) ) ) )
6665impcom 120 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 101    /\ w3a 894   A.wal 1255    = wceq 1257   E.wex 1395    e. wcel 1407   {cab 2040   A.wral 2321   E.wrex 2322   _Vcvv 2572    u. cun 2940   {csn 3400   <.cop 3403   Oncon0 4125    |` cres 4372   Fun wfun 4921    Fn wfn 4922   ` cfv 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-id 4055  df-iord 4128  df-on 4130  df-suc 4133  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-recs 5948
This theorem is referenced by:  tfrlemi14d  5975  tfrexlem  5976
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