Theorem freccl 5993

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 5178	. . . . 5 |- ( x = B -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` B ) )
3	2	eleq1d 2106	. . . 4 |- ( x = B -> ( (frec ( F , A ) ` x ) e. S <-> (frec ( F , A ) ` B ) e. S ) )
4	3	imbi2d 219	. . 3 |- ( x = B -> ( ( ph -> (frec ( F , A ) ` x ) e. S ) <-> ( ph -> (frec ( F , A ) ` B ) e. S ) ) )
5		fveq2 5178	. . . . 5 |- ( x = (/) -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` (/) ) )
6	5	eleq1d 2106	. . . 4 |- ( x = (/) -> ( (frec ( F , A ) ` x ) e. S <-> (frec ( F , A ) ` (/) ) e. S ) )
7		fveq2 5178	. . . . 5 |- ( x = y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` y ) )
8	7	eleq1d 2106	. . . 4 |- ( x = y -> ( (frec ( F , A ) ` x ) e. S <-> (frec ( F , A ) ` y ) e. S ) )
9		fveq2 5178	. . . . 5 |- ( x = suc y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` suc y ) )
10	9	eleq1d 2106	. . . 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 5983	. . . . . 6 |- ( A e. S -> (frec ( F , A ) ` (/) ) = A )
13	11, 12	syl 14	. . . . 5 |- ( ph -> (frec ( F , A ) ` (/) ) = A )
14	13, 11	eqeltrd 2114	. . . 4 |- ( ph -> (frec ( F , A ) ` (/) ) e. S )
15		freccl.ex	. . . . . . . . . 10 |- ( ph -> A. z ( F ` z ) e. _V )
16		frecfnom 5986	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. S ) -> frec ( F , A ) Fn om )
17	15, 11, 16	syl2anc 391	. . . . . . . . 9 |- ( ph -> frec ( F , A ) Fn om )
18		funfvex 5192	. . . . . . . . . 10 |- ( ( Fun frec ( F , A ) /\ y e. dom frec ( F , A ) ) -> (frec ( F , A ) ` y ) e. _V )
19	18	funfni 4999	. . . . . . . . 9 |- ( (frec ( F , A ) Fn om /\ y e. om ) -> (frec ( F , A ) ` y ) e. _V )
20	17, 19	sylan 267	. . . . . . . 8 |- ( ( ph /\ y e. om ) -> (frec ( F , A ) ` y ) e. _V )
21		isset 2561	. . . . . . . 8 |- ( (frec ( F , A ) ` y ) e. _V <-> E. z z = (frec ( F , A ) ` y ) )
22	20, 21	sylib 127	. . . . . . 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 108	. . . . . . . . . . . 12 |- ( ph -> ( z e. S -> ( F ` z ) e. S ) )
25	24	adantr 261	. . . . . . . . . . 11 |- ( ( ph /\ z = (frec ( F , A ) ` y ) ) -> ( z e. S -> ( F ` z ) e. S ) )
26		eleq1 2100	. . . . . . . . . . . 12 |- ( z = (frec ( F , A ) ` y ) -> ( z e. S <-> (frec ( F , A ) ` y ) e. S ) )
27	26	adantl 262	. . . . . . . . . . 11 |- ( ( ph /\ z = (frec ( F , A ) ` y ) ) -> ( z e. S <-> (frec ( F , A ) ` y ) e. S ) )
28		fveq2 5178	. . . . . . . . . . . . 13 |- ( z = (frec ( F , A ) ` y ) -> ( F ` z ) = ( F ` (frec ( F , A ) ` y ) ) )
29	28	eleq1d 2106	. . . . . . . . . . . 12 |- ( z = (frec ( F , A ) ` y ) -> ( ( F ` z ) e. S <-> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
30	29	adantl 262	. . . . . . . . . . 11 |- ( ( ph /\ z = (frec ( F , A ) ` y ) ) -> ( ( F ` z ) e. S <-> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
31	25, 27, 30	3imtr3d 191	. . . . . . . . . 10 |- ( ( ph /\ z = (frec ( F , A ) ` y ) ) -> ( (frec ( F , A ) ` y ) e. S -> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
32	31	ex 108	. . . . . . . . 9 |- ( ph -> ( z = (frec ( F , A ) ` y ) -> ( (frec ( F , A ) ` y ) e. S -> ( F ` (frec ( F , A ) ` y ) ) e. S ) ) )
33	32	exlimdv 1700	. . . . . . . 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 261	. . . . . . 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 261	. . . . . . . 8 |- ( ( ph /\ y e. om ) -> A. z ( F ` z ) e. _V )
37	11	adantr 261	. . . . . . . 8 |- ( ( ph /\ y e. om ) -> A e. S )
38		simpr 103	. . . . . . . 8 |- ( ( ph /\ y e. om ) -> y e. om )
39		frecsuc 5991	. . . . . . . 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 1135	. . . . . . 7 |- ( ( ph /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
41	40	eleq1d 2106	. . . . . 6 |- ( ( ph /\ y e. om ) -> ( (frec ( F , A ) ` suc y ) e. S <-> ( F ` (frec ( F , A ) ` y ) ) e. S ) )
42	35, 41	sylibrd 158	. . . . 5 |- ( ( ph /\ y e. om ) -> ( (frec ( F , A ) ` y ) e. S -> (frec ( F , A ) ` suc y ) e. S ) )
43	42	expcom 109	. . . 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 4324	. . 3 |- ( x e. om -> ( ph -> (frec ( F , A ) ` x ) e. S ) )
45	4, 44	vtoclga 2619	. 2 |- ( B e. om -> ( ph -> (frec ( F , A ) ` B ) e. S ) )
46	1, 45	mpcom 32	1 |- ( ph -> (frec ( F , A ) ` B ) e. S )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   (/)c0 3224   suc csuc 4102   omcom 4313   Fn wfn 4897   ` cfv 4902  freccfrec 5977

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  frecuzrdgrrn  9194