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Theorem frecfnom 6377
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 2170 . . . 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 2170 . . . . 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 6376 . . . 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 6311 . . 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 5310 . . 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 4595 . . . . . 6  |-  om  C_  On
8 sseqin2 3346 . . . . . 6  |-  ( om  C_  On  <->  ( On  i^i  om )  =  om )
97, 8mpbi 144 . . . . 5  |-  ( On 
i^i  om )  =  om
109reseq2i 4886 . . . 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 6367 . . . 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 2194 . . 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 5289 . . 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 424 . 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 121 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 103    <-> wb 104    \/ wo 703   A.wal 1346    = wceq 1348    e. wcel 2141   {cab 2156   E.wrex 2449   _Vcvv 2730    i^i cin 3120    C_ wss 3121   (/)c0 3414    |-> cmpt 4048   Oncon0 4346   suc csuc 4348   omcom 4572   dom cdm 4609    |` cres 4611    Fn wfn 5191   ` cfv 5196  recscrecs 6280  freccfrec 6366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-recs 6281  df-frec 6367
This theorem is referenced by:  frecrdg  6384  frec2uzrand  10348  frec2uzf1od  10349  frecfzennn  10369  hashinfom  10699
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