Theorem freccl 6016

Description: Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)

Hypotheses

Ref	Expression
freccl.ex	|- ( ph -> A. z ( F ` z ) e. _V )
freccl.a	|- ( ph -> A e. S )
freccl.cl	|- ( ( ph /\ z e. S ) -> ( F ` z ) e. S )
freccl.b	|- ( ph -> B e. om )

Assertion

Ref	Expression
freccl	|- ( ph -> (frec ( F , A ) ` B ) e. S )

Distinct variable groups:   z, A   z, F   z, S   ph, z

Allowed substitution hint:   B( z)

Proof of Theorem freccl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		freccl.b	. 2 |- ( ph -> B e. om )
2		fveq2 5198	. . . . 5 |- ( x = B -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` B ) )
3	2	eleq1d 2147	. . . 4 |- ( x = B -> ( (frec ( F , A ) ` x ) e. S <-> (frec ( F , A ) ` B ) e. S ) )
4	3	imbi2d 228	. . 3 |- ( x = B -> ( ( ph -> (frec ( F , A ) ` x ) e. S ) <-> ( ph -> (frec ( F , A ) ` B ) e. S ) ) )
5		fveq2 5198	. . . . 5 |- ( x = (/) -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` (/) ) )
6	5	eleq1d 2147	. . . 4 |- ( x = (/) -> ( (frec ( F , A ) ` x ) e. S <-> (frec ( F , A ) ` (/) ) e. S ) )
7		fveq2 5198	. . . . 5 |- ( x = y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` y ) )
8	7	eleq1d 2147	. . . 4 |- ( x = y -> ( (frec ( F , A ) ` x ) e. S <-> (frec ( F , A ) ` y ) e. S ) )
9		fveq2 5198	. . . . 5 |- ( x = suc y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` suc y ) )
10	9	eleq1d 2147	. . . 4 |- ( x = suc y -> ( (frec ( F , A ) ` x ) e. S <-> (frec ( F , A ) ` suc y ) e. S ) )
11		freccl.a	. . . . . 6 |- ( ph -> A e. S )
12		frec0g 6006	. . . . . 6 |- ( A e. S -> (frec ( F , A ) ` (/) ) = A )
13	11, 12	syl 14	. . . . 5 |- ( ph -> (frec ( F , A ) ` (/) ) = A )
14	13, 11	eqeltrd 2155	. . . 4 |- ( ph -> (frec ( F , A ) ` (/) ) e. S )
15		freccl.ex	. . . . . . . . . 10 |- ( ph -> A. z ( F ` z ) e. _V )
16		frecfnom 6009	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. S ) -> frec ( F , A ) Fn om )
17	15, 11, 16	syl2anc 403	. . . . . . . . 9 |- ( ph -> frec ( F , A ) Fn om )
18		funfvex 5212	. . . . . . . . . 10 |- ( ( Fun frec ( F , A ) /\ y e. dom frec ( F , A ) ) -> (frec ( F , A ) ` y ) e. _V )
19	18	funfni 5019	. . . . . . . . 9 |- ( (frec ( F , A ) Fn om /\ y e. om ) -> (frec ( F , A ) ` y ) e. _V )
20	17, 19	sylan 277	. . . . . . . 8 |- ( ( ph /\ y e. om ) -> (frec ( F , A ) ` y ) e. _V )
21		isset 2605	. . . . . . . 8 |- ( (frec ( F , A ) ` y ) e. _V <-> E. z z = (frec ( F , A ) ` y ) )
22	20, 21	sylib 120	. . . . . . 7 |- ( ( ph /\ y e. om ) -> E. z z = (frec ( F , A ) ` y ) )
23		freccl.cl	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. S ) -> ( F ` z ) e. S )
24	23	ex 113	. . . . . . . . . . . 12 |- ( ph -> ( z e. S -> ( F ` z ) e. S ) )
25	24	adantr 270	. . . . . . . . . . 11 |- ( ( ph /\ z = (frec ( F , A ) ` y ) ) -> ( z e. S -> ( F ` z ) e. S ) )
26		eleq1 2141	. . . . . . . . . . . 12 |- ( z = (frec ( F , A ) ` y ) -> ( z e. S <-> (frec ( F , A ) ` y ) e. S ) )
27	26	adantl 271	. . . . . . . . . . 11 |- ( ( ph /\ z = (frec ( F , A ) ` y ) ) -> ( z e. S <-> (frec ( F , A ) ` y ) e. S ) )
28		fveq2 5198	. . . . . . . . . . . . 13 |- ( z = (frec ( F , A ) ` y ) -> ( F ` z ) = ( F ` (frec ( F , A ) ` y ) ) )
29	28	eleq1d 2147	. . . . . . . . . . . 12 |- ( z = (frec ( F , A ) ` y ) -> ( ( F ` z ) e. S <-> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
30	29	adantl 271	. . . . . . . . . . 11 |- ( ( ph /\ z = (frec ( F , A ) ` y ) ) -> ( ( F ` z ) e. S <-> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
31	25, 27, 30	3imtr3d 200	. . . . . . . . . 10 |- ( ( ph /\ z = (frec ( F , A ) ` y ) ) -> ( (frec ( F , A ) ` y ) e. S -> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
32	31	ex 113	. . . . . . . . 9 |- ( ph -> ( z = (frec ( F , A ) ` y ) -> ( (frec ( F , A ) ` y ) e. S -> ( F ` (frec ( F , A ) ` y ) ) e. S ) ) )
33	32	exlimdv 1740	. . . . . . . 8 |- ( ph -> ( E. z z = (frec ( F , A ) ` y ) -> ( (frec ( F , A ) ` y ) e. S -> ( F ` (frec ( F , A ) ` y ) ) e. S ) ) )
34	33	adantr 270	. . . . . . 7 |- ( ( ph /\ y e. om ) -> ( E. z z = (frec ( F , A ) ` y ) -> ( (frec ( F , A ) ` y ) e. S -> ( F ` (frec ( F , A ) ` y ) ) e. S ) ) )
35	22, 34	mpd 13	. . . . . 6 |- ( ( ph /\ y e. om ) -> ( (frec ( F , A ) ` y ) e. S -> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
36	15	adantr 270	. . . . . . . 8 |- ( ( ph /\ y e. om ) -> A. z ( F ` z ) e. _V )
37	11	adantr 270	. . . . . . . 8 |- ( ( ph /\ y e. om ) -> A e. S )
38		simpr 108	. . . . . . . 8 |- ( ( ph /\ y e. om ) -> y e. om )
39		frecsuc 6014	. . . . . . . 8 |- ( ( A. z ( F ` z ) e. _V /\ A e. S /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
40	36, 37, 38, 39	syl3anc 1169	. . . . . . 7 |- ( ( ph /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
41	40	eleq1d 2147	. . . . . 6 |- ( ( ph /\ y e. om ) -> ( (frec ( F , A ) ` suc y ) e. S <-> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
42	35, 41	sylibrd 167	. . . . 5 |- ( ( ph /\ y e. om ) -> ( (frec ( F , A ) ` y ) e. S -> (frec ( F , A ) ` suc y ) e. S ) )
43	42	expcom 114	. . . 4 |- ( y e. om -> ( ph -> ( (frec ( F , A ) ` y ) e. S -> (frec ( F , A ) ` suc y ) e. S ) ) )
44	6, 8, 10, 14, 43	finds2 4342	. . 3 |- ( x e. om -> ( ph -> (frec ( F , A ) ` x ) e. S ) )
45	4, 44	vtoclga 2664	. 2 |- ( B e. om -> ( ph -> (frec ( F , A ) ` B ) e. S ) )
46	1, 45	mpcom 36	1 |- ( ph -> (frec ( F , A ) ` B ) e. S )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   (/)c0 3251   suc csuc 4120   omcom 4331   Fn wfn 4917   ` cfv 4922  freccfrec 6000

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-frec 6001

This theorem is referenced by:  frecuzrdgrrn  9410