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Theorem frecfun 6260
Description: Finite recursion produces a function. See also frecfnom 6266 which also states that the domain of that function is  om but which puts conditions on  A and  F. (Contributed by Jim Kingdon, 13-Feb-2022.)
Assertion
Ref Expression
frecfun  |-  Fun frec ( F ,  A )

Proof of Theorem frecfun
Dummy variables  g  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrfun 6185 . . 3  |-  Fun recs (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )
2 funres 5134 . . 3  |-  ( Fun recs
( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  ->  Fun  (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
)
31, 2ax-mp 5 . 2  |-  Fun  (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
4 df-frec 6256 . . 3  |- 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 )
54funeqi 5114 . 2  |-  ( Fun frec
( F ,  A
)  <->  Fun  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
)
63, 5mpbir 145 1  |-  Fun frec ( F ,  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 682    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660   (/)c0 3333    |-> cmpt 3959   suc csuc 4257   omcom 4474   dom cdm 4509    |` cres 4511   Fun wfun 5087   ` cfv 5093  recscrecs 6169  freccfrec 6255
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-recs 6170  df-frec 6256
This theorem is referenced by: (None)
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