Theorem frecuzrdgdomlem 10837

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 10835	. . . . 5 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
7		frn 5540	. . . . 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 4978	. . . 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 5216	. . 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 5534	. . . . . . . . . . . 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 10826	. . . . . . . . . . . 12 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
19		f1ocnvdm 5981	. . . . . . . . . . . 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 5537	. . . . . . . . . . . . 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 5833	. . . . . . . . . 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 6403	. . . . . . . 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 10836	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = ( G ` ( `' G ` v ) ) )
35		f1ocnvfv2 5978	. . . . . . . . . 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 3908	. . . . . . 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 6394	. . . . . . 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 4984	. . . . . 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 3711   |-> cmpt 4190   omcom 4735   X. cxp 4770   `'ccnv 4771   dom cdm 4772   ran crn 4773   Fun wfun 5369   -->wf 5371   -1-1-onto->wf1o 5374   ` cfv 5375  (class class class)co 6079   e. cmpo 6081   1stc1st 6366   2ndc2nd 6367  freccfrec 6655   1c1 8174   + caddc 8176   ZZcz 9627   ZZ>=cuz 9904

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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-ltadd 8289

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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905

This theorem is used by:  frecuzrdgdom  10838