Theorem frecuzrdgdomlem 10437

Description: The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)

Hypotheses

Ref	Expression
frecuzrdgrclt.c	|- ( ph -> C e. ZZ )
frecuzrdgrclt.a	|- ( ph -> A e. S )
frecuzrdgrclt.t	|- ( ph -> S C_ T )
frecuzrdgrclt.f	|- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
frecuzrdgrclt.r	|- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
frecuzrdgdomlem.g	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )

Assertion

Ref	Expression
frecuzrdgdomlem	|- ( ph -> dom ran R = ( ZZ>= ` C ) )

Distinct variable groups:   x, C, y   x, F, y   x, S, y   x, T, y   ph, x, y   x, R, y   x, G, y

Allowed substitution hints:   A( x, y)

Proof of Theorem frecuzrdgdomlem

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecuzrdgrclt.c	. . . . . 6 |- ( ph -> C e. ZZ )
2		frecuzrdgrclt.a	. . . . . 6 |- ( ph -> A e. S )
3		frecuzrdgrclt.t	. . . . . 6 |- ( ph -> S C_ T )
4		frecuzrdgrclt.f	. . . . . 6 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
5		frecuzrdgrclt.r	. . . . . 6 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
6	1, 2, 3, 4, 5	frecuzrdgrclt 10435	. . . . 5 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
7		frn 5390	. . . . 5 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> ran R C_ ( ( ZZ>= ` C ) X. S ) )
8	6, 7	syl 14	. . . 4 |- ( ph -> ran R C_ ( ( ZZ>= ` C ) X. S ) )
9		dmss 4841	. . . 4 |- ( ran R C_ ( ( ZZ>= ` C ) X. S ) -> dom ran R C_ dom ( ( ZZ>= ` C ) X. S ) )
10	8, 9	syl 14	. . 3 |- ( ph -> dom ran R C_ dom ( ( ZZ>= ` C ) X. S ) )
11		dmxpss 5074	. . 3 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
12	10, 11	sstrdi 3182	. 2 |- ( ph -> dom ran R C_ ( ZZ>= ` C ) )
13	8	adantr 276	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ran R C_ ( ( ZZ>= ` C ) X. S ) )
14		ffun 5384	. . . . . . . . . . . 12 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> Fun R )
15	6, 14	syl 14	. . . . . . . . . . 11 |- ( ph -> Fun R )
16	15	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> Fun R )
17		frecuzrdgdomlem.g	. . . . . . . . . . . . 13 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
18	1, 17	frec2uzf1od 10426	. . . . . . . . . . . 12 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
19		f1ocnvdm 5799	. . . . . . . . . . . 12 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
20	18, 19	sylan 283	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. om )
21		fdm 5387	. . . . . . . . . . . . 13 |- ( R : om --> ( ( ZZ>= ` C ) X. S ) -> dom R = om )
22	6, 21	syl 14	. . . . . . . . . . . 12 |- ( ph -> dom R = om )
23	22	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> dom R = om )
24	20, 23	eleqtrrd 2269	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. dom R )
25		fvelrn 5664	. . . . . . . . . 10 |- ( ( Fun R /\ ( `' G ` v ) e. dom R ) -> ( R ` ( `' G ` v ) ) e. ran R )
26	16, 24, 25	syl2anc 411	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ran R )
27	13, 26	sseldd 3171	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		1st2nd2 6195	. . . . . . . 8 |- ( ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( R ` ( `' G ` v ) ) = <. ( 1st ` ( R ` ( `' G ` v ) ) ) , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
29	27, 28	syl 14	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) = <. ( 1st ` ( R ` ( `' G ` v ) ) ) , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
30	1	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> C e. ZZ )
31	2	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> A e. S )
32	3	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> S C_ T )
33	4	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ v e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
34	30, 31, 32, 33, 5, 20, 17	frecuzrdgg 10436	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = ( G ` ( `' G ` v ) ) )
35		f1ocnvfv2 5796	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ v e. ( ZZ>= ` C ) ) -> ( G ` ( `' G ` v ) ) = v )
36	18, 35	sylan 283	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( G ` ( `' G ` v ) ) = v )
37	34, 36	eqtrd 2222	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = v )
38	37	opeq1d 3799	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. ( 1st ` ( R ` ( `' G ` v ) ) ) , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. = <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
39	29, 38	eqtrd 2222	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) = <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
40	39, 26	eqeltrrd 2267	. . . . 5 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R )
41		simpr 110	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. ( ZZ>= ` C ) )
42		xp2nd 6186	. . . . . . 7 |- ( ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S )
43	27, 42	syl 14	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S )
44		opeldmg 4847	. . . . . 6 |- ( ( v e. ( ZZ>= ` C ) /\ ( 2nd ` ( R ` ( `' G ` v ) ) ) e. S ) -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R -> v e. dom ran R ) )
45	41, 43, 44	syl2anc 411	. . . . 5 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. e. ran R -> v e. dom ran R ) )
46	40, 45	mpd 13	. . . 4 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> v e. dom ran R )
47	46	ex 115	. . 3 |- ( ph -> ( v e. ( ZZ>= ` C ) -> v e. dom ran R ) )
48	47	ssrdv 3176	. 2 |- ( ph -> ( ZZ>= ` C ) C_ dom ran R )
49	12, 48	eqssd 3187	1 |- ( ph -> dom ran R = ( ZZ>= ` C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   C_ wss 3144   <.cop 3610   |-> cmpt 4079   omcom 4604   X. cxp 4639   `'ccnv 4640   dom cdm 4641   ran crn 4642   Fun wfun 5226   -->wf 5228   -1-1-onto->wf1o 5231   ` cfv 5232  (class class class)co 5892   e. cmpo 5894   1stc1st 6158   2ndc2nd 6159  freccfrec 6410   1c1 7832   + caddc 7834   ZZcz 9273   ZZ>=cuz 9548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-addcom 7931  ax-addass 7933  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-0id 7939  ax-rnegex 7940  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-ltadd 7947

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-inn 8940  df-n0 9197  df-z 9274  df-uz 9549

This theorem is referenced by:  frecuzrdgdom  10438