Theorem frecfun 6244

Description: Finite recursion produces a function. See also frecfnom 6250 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.

Step	Hyp	Ref	Expression
1		tfrfun 6169	. . 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 5120	. . 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 ) )
3	1, 2	ax-mp 7	. 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 6240	. . 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 )
5	4	funeqi 5100	. 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 ) )
6	3, 5	mpbir 145	1 |- Fun frec ( F , A )

Syntax hints:   /\ wa 103   \/ wo 680   = wceq 1312   e. wcel 1461   {cab 2099   E.wrex 2389   _Vcvv 2655   (/)c0 3327   |-> cmpt 3947   suc csuc 4245   omcom 4462   dom cdm 4497   |` cres 4499   Fun wfun 5073   ` cfv 5079  recscrecs 6153  freccfrec 6239

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-setind 4410

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-res 4509  df-iota 5044  df-fun 5081  df-fn 5082  df-fv 5087  df-recs 6154  df-frec 6240

This theorem is referenced by: (None)