Theorem frecfnom 6120

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 2085	. . . 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 2085	. . . . 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 6119	. . . 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 6054	. . 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 5093	. . 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 4400	. . . . . 6 |- om C_ On
8		sseqin2 3208	. . . . . 6 |- ( om C_ On <-> ( On i^i om ) = om )
9	7, 8	mpbi 143	. . . . 5 |- ( On i^i om ) = om
10	9	reseq2i 4678	. . . 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 6110	. . . 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 2108	. . 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 5072	. . 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 417	. 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 120	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> frec ( F , A ) Fn om )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 662   A.wal 1285   = wceq 1287   e. wcel 1436   {cab 2071   E.wrex 2356   _Vcvv 2615   i^i cin 2987   C_ wss 2988   (/)c0 3275   |-> cmpt 3874   Oncon0 4164   suc csuc 4166   omcom 4378   dom cdm 4411   |` cres 4413   Fn wfn 4976   ` cfv 4981  recscrecs 6023  freccfrec 6109

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-recs 6024  df-frec 6110

This theorem is referenced by:  frecrdg  6127  frec2uzrand  9740  frec2uzf1od  9741  frecfzennn  9761  hashinfom  10083