Theorem frecuzrdgdomlem 10742

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 10740	. . . . 5 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
7		frn 5498	. . . . 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 4936	. . . 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 5174	. . 3 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
12	10, 11	sstrdi 3240	. 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 5492	. . . . . . . . . . . 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 10731	. . . . . . . . . . . 12 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
19		f1ocnvdm 5932	. . . . . . . . . . . 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 5495	. . . . . . . . . . . . 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 2311	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. dom R )
25		fvelrn 5786	. . . . . . . . . 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 3229	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		1st2nd2 6347	. . . . . . . 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 10741	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = ( G ` ( `' G ` v ) ) )
35		f1ocnvfv2 5929	. . . . . . . . . 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 2264	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = v )
38	37	opeq1d 3873	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. ( 1st ` ( R ` ( `' G ` v ) ) ) , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. = <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
39	29, 38	eqtrd 2264	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) = <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
40	39, 26	eqeltrrd 2309	. . . . 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 6338	. . . . . . 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 4942	. . . . . 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 3234	. 2 |- ( ph -> ( ZZ>= ` C ) C_ dom ran R )
49	12, 48	eqssd 3245	1 |- ( ph -> dom ran R = ( ZZ>= ` C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   C_ wss 3201   <.cop 3676   |-> cmpt 4155   omcom 4694   X. cxp 4729   `'ccnv 4730   dom cdm 4731   ran crn 4732   Fun wfun 5327   -->wf 5329   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   1stc1st 6310   2ndc2nd 6311  freccfrec 6599   1c1 8093   + caddc 8095   ZZcz 9540   ZZ>=cuz 9816

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817

This theorem is referenced by:  frecuzrdgdom  10743