Theorem frecuzrdgcl 9199

Description: Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 9185 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 31-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	|- ( ph -> C e. ZZ )
frec2uz.2	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
uzrdg.s	|- ( ph -> S e. V )
uzrdg.a	|- ( ph -> A e. S )
uzrdg.f	|- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
uzrdg.2	|- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
frecuzrdgfn.3	|- ( ph -> T = ran R )

Assertion

Ref	Expression
frecuzrdgcl	|- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` B ) e. S )

Distinct variable groups:   y, A   x, C, y   y, G   x, F, y   x, S, y   ph, x, y   x, B, y

Allowed substitution hints:   A( x)   R( x, y)   T( x, y)   G( x)   V( x, y)

Proof of Theorem frecuzrdgcl

Step	Hyp	Ref	Expression
1		frec2uz.1	. . . . . 6 |- ( ph -> C e. ZZ )
2	1	adantr 261	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> C e. ZZ )
3		frec2uz.2	. . . . 5 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
4		uzrdg.s	. . . . . 6 |- ( ph -> S e. V )
5	4	adantr 261	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> S e. V )
6		uzrdg.a	. . . . . 6 |- ( ph -> A e. S )
7	6	adantr 261	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> A e. S )
8		uzrdg.f	. . . . . 6 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
9	8	adantlr 446	. . . . 5 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
10		uzrdg.2	. . . . 5 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
11		simpr 103	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> B e. ( ZZ>= ` C ) )
12	2, 3, 5, 7, 9, 10, 11	frecuzrdglem 9197	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. ran R )
13		frecuzrdgfn.3	. . . . 5 |- ( ph -> T = ran R )
14	13	adantr 261	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> T = ran R )
15	12, 14	eleqtrrd 2117	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T )
16	1, 3, 4, 6, 8, 10, 13	frecuzrdgfn 9198	. . . . 5 |- ( ph -> T Fn ( ZZ>= ` C ) )
17		fnfun 4996	. . . . 5 |- ( T Fn ( ZZ>= ` C ) -> Fun T )
18		funopfv 5213	. . . . 5 |- ( Fun T -> ( <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T -> ( T ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
19	16, 17, 18	3syl 17	. . . 4 |- ( ph -> ( <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T -> ( T ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
20	19	adantr 261	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T -> ( T ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
21	15, 20	mpd 13	. 2 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) )
22	1, 3	frec2uzf1od 9192	. . . . 5 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
23		f1ocnvdm 5421	. . . . 5 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
24	22, 23	sylan 267	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
25	1, 3, 4, 6, 8, 10	frecuzrdgrrn 9194	. . . 4 |- ( ( ph /\ ( `' G ` B ) e. om ) -> ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) )
26	24, 25	syldan 266	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) )
27		xp2nd 5793	. . 3 |- ( ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) e. S )
28	26, 27	syl 14	. 2 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) e. S )
29	21, 28	eqeltrd 2114	1 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` B ) e. S )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   <.cop 3378   |-> cmpt 3818   omcom 4313   X. cxp 4343   `'ccnv 4344   ran crn 4346   Fun wfun 4896   Fn wfn 4897   -1-1-onto->wf1o 4901   ` cfv 4902  (class class class)co 5512   |-> cmpt2 5514   2ndc2nd 5766  freccfrec 5977   1c1 6890   + caddc 6892   ZZcz 8245   ZZ>=cuz 8473

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  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  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-nel 2207  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-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  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-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474

This theorem is referenced by:  iseqcl  9223