Theorem frecfun 6393

Description: Finite recursion produces a function. See also frecfnom 6399 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 6318	. . 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 5256	. . 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 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 6389	. . 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 5236	. 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 146	1 |- Fun frec ( F , A )

Syntax hints:   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2737   (/)c0 3422   |-> cmpt 4063   suc csuc 4364   omcom 4588   dom cdm 4625   |` cres 4627   Fun wfun 5209   ` cfv 5215  recscrecs 6302  freccfrec 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-setind 4535

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-iord 4365  df-on 4367  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-recs 6303  df-frec 6389

This theorem is referenced by: (None)