Theorem frecuzrdgdomlem 10639

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	⊢ (𝜑 → 𝐶 ∈ ℤ)
frecuzrdgrclt.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
frecuzrdgrclt.t	⊢ (𝜑 → 𝑆 ⊆ 𝑇)
frecuzrdgrclt.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrclt.r	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgdomlem.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)

Assertion

Ref	Expression
frecuzrdgdomlem	⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))

Distinct variable groups:   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝜑,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 ⊢ (𝜑 → 𝐶 ∈ ℤ)
2		frecuzrdgrclt.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		frecuzrdgrclt.t	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
4		frecuzrdgrclt.f	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
5		frecuzrdgrclt.r	. . . . . 6 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
6	1, 2, 3, 4, 5	frecuzrdgrclt 10637	. . . . 5 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
7		frn 5482	. . . . 5 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
8	6, 7	syl 14	. . . 4 ⊢ (𝜑 → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
9		dmss 4922	. . . 4 ⊢ (ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆) → dom ran 𝑅 ⊆ dom ((ℤ≥‘𝐶) × 𝑆))
10	8, 9	syl 14	. . 3 ⊢ (𝜑 → dom ran 𝑅 ⊆ dom ((ℤ≥‘𝐶) × 𝑆))
11		dmxpss 5159	. . 3 ⊢ dom ((ℤ≥‘𝐶) × 𝑆) ⊆ (ℤ≥‘𝐶)
12	10, 11	sstrdi 3236	. 2 ⊢ (𝜑 → dom ran 𝑅 ⊆ (ℤ≥‘𝐶))
13	8	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
14		ffun 5476	. . . . . . . . . . . 12 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → Fun 𝑅)
15	6, 14	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝑅)
16	15	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → Fun 𝑅)
17		frecuzrdgdomlem.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
18	1, 17	frec2uzf1od 10628	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
19		f1ocnvdm 5905	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
20	18, 19	sylan 283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
21		fdm 5479	. . . . . . . . . . . . 13 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → dom 𝑅 = ω)
22	6, 21	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑅 = ω)
23	22	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → dom 𝑅 = ω)
24	20, 23	eleqtrrd 2309	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ dom 𝑅)
25		fvelrn 5766	. . . . . . . . . 10 ⊢ ((Fun 𝑅 ∧ (◡𝐺‘𝑣) ∈ dom 𝑅) → (𝑅‘(◡𝐺‘𝑣)) ∈ ran 𝑅)
26	16, 24, 25	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ran 𝑅)
27	13, 26	sseldd 3225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
28		1st2nd2 6321	. . . . . . . 8 ⊢ ((𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (𝑅‘(◡𝐺‘𝑣)) = ⟨(1st ‘(𝑅‘(◡𝐺‘𝑣))), (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
29	27, 28	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) = ⟨(1st ‘(𝑅‘(◡𝐺‘𝑣))), (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
30	1	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ)
31	2	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝐴 ∈ 𝑆)
32	3	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑆 ⊆ 𝑇)
33	4	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
34	30, 31, 32, 33, 5, 20, 17	frecuzrdgg 10638	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝑣))) = (𝐺‘(◡𝐺‘𝑣)))
35		f1ocnvfv2 5902	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝑣)) = 𝑣)
36	18, 35	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝑣)) = 𝑣)
37	34, 36	eqtrd 2262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝑣))) = 𝑣)
38	37	opeq1d 3863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨(1st ‘(𝑅‘(◡𝐺‘𝑣))), (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ = ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
39	29, 38	eqtrd 2262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) = ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
40	39, 26	eqeltrrd 2307	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
41		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑣 ∈ (ℤ≥‘𝐶))
42		xp2nd 6312	. . . . . . 7 ⊢ ((𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
43	27, 42	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
44		opeldmg 4928	. . . . . 6 ⊢ ((𝑣 ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆) → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅 → 𝑣 ∈ dom ran 𝑅))
45	41, 43, 44	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅 → 𝑣 ∈ dom ran 𝑅))
46	40, 45	mpd 13	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑣 ∈ dom ran 𝑅)
47	46	ex 115	. . 3 ⊢ (𝜑 → (𝑣 ∈ (ℤ≥‘𝐶) → 𝑣 ∈ dom ran 𝑅))
48	47	ssrdv 3230	. 2 ⊢ (𝜑 → (ℤ≥‘𝐶) ⊆ dom ran 𝑅)
49	12, 48	eqssd 3241	1 ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ⊆ wss 3197  ⟨cop 3669   ↦ cmpt 4145  ωcom 4682   × cxp 4717  ^◡ccnv 4718  dom cdm 4719  ran crn 4720  Fun wfun 5312  ⟶wf 5314  –1-1-onto→wf1o 5317  ‘cfv 5318  (class class class)co 6001   ∈ cmpo 6003  1^st c1st 6284  2^nd c2nd 6285  freccfrec 6536  1c1 8000   + caddc 8002  ℤcz 9446  ℤ_≥cuz 9722

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723

This theorem is referenced by:  frecuzrdgdom  10640