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Theorem frecfnom 6017
Description: The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
Assertion
Ref Expression
frecfnom  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  -> frec ( F ,  A )  Fn 
om )
Distinct variable groups:    z, A    z, F
Allowed substitution hint:    V( z)

Proof of Theorem frecfnom
Dummy variables  g  m  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2056 . . . 4  |- recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  = recs (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )
2 eqid 2056 . . . . 5  |-  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
32frectfr 6016 . . . 4  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  A. y
( Fun  ( g  e.  _V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  /\  ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  y )  e.  _V ) )
41, 3tfri1d 5980 . . 3  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  -> recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  Fn  On )
5 fnresin1 5041 . . 3  |-  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  Fn  On  ->  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  ( On  i^i  om ) )  Fn  ( On  i^i  om ) )
64, 5syl 14 . 2  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  ( On  i^i  om ) )  Fn  ( On  i^i  om ) )
7 omsson 4363 . . . . . 6  |-  om  C_  On
8 sseqin2 3184 . . . . . 6  |-  ( om  C_  On  <->  ( On  i^i  om )  =  om )
97, 8mpbi 137 . . . . 5  |-  ( On 
i^i  om )  =  om
109reseq2i 4637 . . . 4  |-  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  ( On  i^i  om ) )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
11 df-frec 6009 . . . 4  |- frec ( F ,  A )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
1210, 11eqtr4i 2079 . . 3  |-  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  ( On  i^i  om ) )  = frec ( F ,  A )
13 fneq12 5020 . . 3  |-  ( ( (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  ( On  i^i  om ) )  = frec ( F ,  A )  /\  ( On  i^i  om )  =  om )  ->  (
(recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  ( On  i^i  om ) )  Fn  ( On  i^i  om )  <-> frec ( F ,  A
)  Fn  om )
)
1412, 9, 13mp2an 410 . 2  |-  ( (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  ( On  i^i  om ) )  Fn  ( On  i^i  om )  <-> frec ( F ,  A
)  Fn  om )
156, 14sylib 131 1  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  -> frec ( F ,  A )  Fn 
om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639   A.wal 1257    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574    i^i cin 2944    C_ wss 2945   (/)c0 3252    |-> cmpt 3846   Oncon0 4128   suc csuc 4130   omcom 4341   dom cdm 4373    |` cres 4375    Fn wfn 4925   ` cfv 4930  recscrecs 5950  freccfrec 6008
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-recs 5951  df-frec 6009
This theorem is referenced by:  frecrdg  6023  freccl  6024  frec2uzrand  9355  frec2uzf1od  9356  frecuzrdgrom  9360  frecfzennn  9367
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