Theorem frecfnom 6562

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.

Step	Hyp	Ref	Expression
1		eqid 2229	. . . 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 2229	. . . . 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 ) ) } )
3	2	frectfr 6561	. . . 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 ) )
4	1, 3	tfri1d 6496	. . 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 5444	. . 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 ) )
6	4, 5	syl 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 4709	. . . . . 6 |- om C_ On
8		sseqin2 3424	. . . . . 6 |- ( om C_ On <-> ( On i^i om ) = om )
9	7, 8	mpbi 145	. . . . 5 |- ( On i^i om ) = om
10	9	reseq2i 5008	. . . 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 6552	. . . 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 )
12	10, 11	eqtr4i 2253	. . 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 5420	. . 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 ) )
14	12, 9, 13	mp2an 426	. 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 )
15	6, 14	sylib 122	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> frec ( F , A ) Fn om )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   A.wal 1393   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2800   i^i cin 3197   C_ wss 3198   (/)c0 3492   |-> cmpt 4148   Oncon0 4458   suc csuc 4460   omcom 4686   dom cdm 4723   |` cres 4725   Fn wfn 5319   ` cfv 5324  recscrecs 6465  freccfrec 6551

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-recs 6466  df-frec 6552

This theorem is referenced by:  frecrdg  6569  frec2uzrand  10657  frec2uzf1od  10658  frecfzennn  10678  hashinfom  11030