Theorem frecuzrdgdomlem 10875

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 10873	. . . . 5 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
7		frn 5545	. . . . 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 4982	. . . 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 5221	. . 3 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
12	10, 11	sstrdi 3260	. 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 5539	. . . . . . . . . . . 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 10864	. . . . . . . . . . . 12 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
19		f1ocnvdm 5990	. . . . . . . . . . . 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 5542	. . . . . . . . . . . . 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 2318	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. dom R )
25		fvelrn 5842	. . . . . . . . . 10 |- ( ( Fun R /\ ( `' G ` v ) e. dom R ) -> ( R ` ( `' G ` v ) ) e. ran R )
26	16, 24, 25	syl2anc 415	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ran R )
27	13, 26	sseldd 3249	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		1st2nd2 6413	. . . . . . . 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 481	. . . . . . . . . 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 10874	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = ( G ` ( `' G ` v ) ) )
35		f1ocnvfv2 5987	. . . . . . . . . 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 2271	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = v )
38	37	opeq1d 3910	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. ( 1st ` ( R ` ( `' G ` v ) ) ) , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. = <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
39	29, 38	eqtrd 2271	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) = <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
40	39, 26	eqeltrrd 2316	. . . . 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 6404	. . . . . . 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 4988	. . . . . 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 415	. . . . 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 3254	. 2 |- ( ph -> ( ZZ>= ` C ) C_ dom ran R )
49	12, 48	eqssd 3265	1 |- ( ph -> dom ran R = ( ZZ>= ` C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   <.cop 3712   |-> cmpt 4193   omcom 4739   X. cxp 4774   `'ccnv 4775   dom cdm 4776   ran crn 4777   Fun wfun 5374   -->wf 5376   -1-1-onto->wf1o 5379   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   1stc1st 6376   2ndc2nd 6377  freccfrec 6665   1c1 8186   + caddc 8188   ZZcz 9655   ZZ>=cuz 9937

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938

This theorem is used by:  frecuzrdgdom  10876