Theorem frecuzrdgdomlem 10599

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 10597	. . . . 5 |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
7		frn 5454	. . . . 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 4896	. . . 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 5132	. . 3 |- dom ( ( ZZ>= ` C ) X. S ) C_ ( ZZ>= ` C )
12	10, 11	sstrdi 3213	. 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 5448	. . . . . . . . . . . 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 10588	. . . . . . . . . . . 12 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
19		f1ocnvdm 5873	. . . . . . . . . . . 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 5451	. . . . . . . . . . . . 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 2287	. . . . . . . . . 10 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( `' G ` v ) e. dom R )
25		fvelrn 5734	. . . . . . . . . 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 3202	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) e. ( ( ZZ>= ` C ) X. S ) )
28		1st2nd2 6284	. . . . . . . 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 10598	. . . . . . . . 9 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = ( G ` ( `' G ` v ) ) )
35		f1ocnvfv2 5870	. . . . . . . . . 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 2240	. . . . . . . 8 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( 1st ` ( R ` ( `' G ` v ) ) ) = v )
38	37	opeq1d 3839	. . . . . . 7 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> <. ( 1st ` ( R ` ( `' G ` v ) ) ) , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. = <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
39	29, 38	eqtrd 2240	. . . . . 6 |- ( ( ph /\ v e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` v ) ) = <. v , ( 2nd ` ( R ` ( `' G ` v ) ) ) >. )
40	39, 26	eqeltrrd 2285	. . . . 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 6275	. . . . . . 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 4902	. . . . . 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 3207	. 2 |- ( ph -> ( ZZ>= ` C ) C_ dom ran R )
49	12, 48	eqssd 3218	1 |- ( ph -> dom ran R = ( ZZ>= ` C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   C_ wss 3174   <.cop 3646   |-> cmpt 4121   omcom 4656   X. cxp 4691   `'ccnv 4692   dom cdm 4693   ran crn 4694   Fun wfun 5284   -->wf 5286   -1-1-onto->wf1o 5289   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   1stc1st 6247   2ndc2nd 6248  freccfrec 6499   1c1 7961   + caddc 7963   ZZcz 9407   ZZ>=cuz 9683

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684

This theorem is referenced by:  frecuzrdgdom  10600