Theorem frecfun 6639

Description: Finite recursion produces a function. See also frecfnom 6645 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 6564	. . 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 5398	. . 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 6635	. . 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 5378	. 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 716   = wceq 1398   e. wcel 2205   {cab 2220   E.wrex 2523   _Vcvv 2815   (/)c0 3512   |-> cmpt 4176   suc csuc 4491   omcom 4717   dom cdm 4754   |` cres 4756   Fun wfun 5351   ` cfv 5357  recscrecs 6548  freccfrec 6634

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-recs 6549  df-frec 6635

This theorem is referenced by: (None)