Theorem frecuzrdgdomlem 10491

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 10489	. . . . 5 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
7		frn 5413	. . . . 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 4862	. . . 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 5097	. . 3 ⊢ dom ((ℤ≥‘𝐶) × 𝑆) ⊆ (ℤ≥‘𝐶)
12	10, 11	sstrdi 3192	. 2 ⊢ (𝜑 → dom ran 𝑅 ⊆ (ℤ≥‘𝐶))
13	8	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
14		ffun 5407	. . . . . . . . . . . 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 10480	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
19		f1ocnvdm 5825	. . . . . . . . . . . 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 5410	. . . . . . . . . . . . 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 2273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ dom 𝑅)
25		fvelrn 5690	. . . . . . . . . 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 3181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
28		1st2nd2 6230	. . . . . . . 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 10490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝑣))) = (𝐺‘(◡𝐺‘𝑣)))
35		f1ocnvfv2 5822	. . . . . . . . . 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 2226	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝑣))) = 𝑣)
38	37	opeq1d 3811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨(1st ‘(𝑅‘(◡𝐺‘𝑣))), (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ = ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
39	29, 38	eqtrd 2226	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) = ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
40	39, 26	eqeltrrd 2271	. . . . 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 6221	. . . . . . 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 4868	. . . . . 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 3186	. 2 ⊢ (𝜑 → (ℤ≥‘𝐶) ⊆ dom ran 𝑅)
49	12, 48	eqssd 3197	1 ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164   ⊆ wss 3154  ⟨cop 3622   ↦ cmpt 4091  ωcom 4623   × cxp 4658  ^◡ccnv 4659  dom cdm 4660  ran crn 4661  Fun wfun 5249  ⟶wf 5251  –1-1-onto→wf1o 5254  ‘cfv 5255  (class class class)co 5919   ∈ cmpo 5921  1^st c1st 6193  2^nd c2nd 6194  freccfrec 6445  1c1 7875   + caddc 7877  ℤcz 9320  ℤ_≥cuz 9595

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596

This theorem is referenced by:  frecuzrdgdom  10492