Theorem frecuzrdgdomlem 10352

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 10350	. . . . 5 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
7		frn 5346	. . . . 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 4803	. . . 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 5034	. . 3 ⊢ dom ((ℤ≥‘𝐶) × 𝑆) ⊆ (ℤ≥‘𝐶)
12	10, 11	sstrdi 3154	. 2 ⊢ (𝜑 → dom ran 𝑅 ⊆ (ℤ≥‘𝐶))
13	8	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
14		ffun 5340	. . . . . . . . . . . 12 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → Fun 𝑅)
15	6, 14	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝑅)
16	15	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → Fun 𝑅)
17		frecuzrdgdomlem.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
18	1, 17	frec2uzf1od 10341	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
19		f1ocnvdm 5749	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
20	18, 19	sylan 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
21		fdm 5343	. . . . . . . . . . . . 13 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → dom 𝑅 = ω)
22	6, 21	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑅 = ω)
23	22	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → dom 𝑅 = ω)
24	20, 23	eleqtrrd 2246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ dom 𝑅)
25		fvelrn 5616	. . . . . . . . . 10 ⊢ ((Fun 𝑅 ∧ (◡𝐺‘𝑣) ∈ dom 𝑅) → (𝑅‘(◡𝐺‘𝑣)) ∈ ran 𝑅)
26	16, 24, 25	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ran 𝑅)
27	13, 26	sseldd 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
28		1st2nd2 6143	. . . . . . . 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 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ)
31	2	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝐴 ∈ 𝑆)
32	3	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑆 ⊆ 𝑇)
33	4	adantlr 469	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
34	30, 31, 32, 33, 5, 20, 17	frecuzrdgg 10351	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝑣))) = (𝐺‘(◡𝐺‘𝑣)))
35		f1ocnvfv2 5746	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝑣)) = 𝑣)
36	18, 35	sylan 281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝑣)) = 𝑣)
37	34, 36	eqtrd 2198	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝑣))) = 𝑣)
38	37	opeq1d 3764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨(1st ‘(𝑅‘(◡𝐺‘𝑣))), (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ = ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
39	29, 38	eqtrd 2198	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) = ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
40	39, 26	eqeltrrd 2244	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
41		simpr 109	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑣 ∈ (ℤ≥‘𝐶))
42		xp2nd 6134	. . . . . . 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 4809	. . . . . 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 409	. . . . 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 114	. . 3 ⊢ (𝜑 → (𝑣 ∈ (ℤ≥‘𝐶) → 𝑣 ∈ dom ran 𝑅))
48	47	ssrdv 3148	. 2 ⊢ (𝜑 → (ℤ≥‘𝐶) ⊆ dom ran 𝑅)
49	12, 48	eqssd 3159	1 ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136   ⊆ wss 3116  ⟨cop 3579   ↦ cmpt 4043  ωcom 4567   × cxp 4602  ^◡ccnv 4603  dom cdm 4604  ran crn 4605  Fun wfun 5182  ⟶wf 5184  –1-1-onto→wf1o 5187  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844  1^st c1st 6106  2^nd c2nd 6107  freccfrec 6358  1c1 7754   + caddc 7756  ℤcz 9191  ℤ_≥cuz 9466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467

This theorem is referenced by:  frecuzrdgdom  10353