Theorem frecuzrdgcl 9415

Description: Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 9401 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 270	. . . . 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 270	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> S e. V )
6		uzrdg.a	. . . . . 6 |- ( ph -> A e. S )
7	6	adantr 270	. . . . 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 460	. . . . 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 108	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> B e. ( ZZ>= ` C ) )
12	2, 3, 5, 7, 9, 10, 11	frecuzrdglem 9413	. . . 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 270	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> T = ran R )
15	12, 14	eleqtrrd 2158	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T )
16	1, 3, 4, 6, 8, 10, 13	frecuzrdgfn 9414	. . . . 5 |- ( ph -> T Fn ( ZZ>= ` C ) )
17		fnfun 5016	. . . . 5 |- ( T Fn ( ZZ>= ` C ) -> Fun T )
18		funopfv 5234	. . . . 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 270	. . 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 9408	. . . . 5 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
23		f1ocnvdm 5441	. . . . 5 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
24	22, 23	sylan 277	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
25	1, 3, 4, 6, 8, 10	frecuzrdgrrn 9410	. . . 4 |- ( ( ph /\ ( `' G ` B ) e. om ) -> ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) )
26	24, 25	syldan 276	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) )
27		xp2nd 5813	. . 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 2155	1 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` B ) e. S )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   <.cop 3401   |-> cmpt 3839   omcom 4331   X. cxp 4361   `'ccnv 4362   ran crn 4364   Fun wfun 4916   Fn wfn 4917   -1-1-onto->wf1o 4921   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   2ndc2nd 5786  freccfrec 6000   1c1 6982   + caddc 6984   ZZcz 8351   ZZ>=cuz 8619

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  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  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-nel 2340  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-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620

This theorem is referenced by:  iseqcl  9443