Theorem frecuzrdgdomlem 10562

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 10560	. . . . 5 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
7		frn 5434	. . . . 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 4877	. . . 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 5113	. . 3 ⊢ dom ((ℤ≥‘𝐶) × 𝑆) ⊆ (ℤ≥‘𝐶)
12	10, 11	sstrdi 3205	. 2 ⊢ (𝜑 → dom ran 𝑅 ⊆ (ℤ≥‘𝐶))
13	8	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
14		ffun 5428	. . . . . . . . . . . 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 10551	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
19		f1ocnvdm 5850	. . . . . . . . . . . 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 5431	. . . . . . . . . . . . 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 2285	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ dom 𝑅)
25		fvelrn 5711	. . . . . . . . . 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 3194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
28		1st2nd2 6261	. . . . . . . 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 10561	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝑣))) = (𝐺‘(◡𝐺‘𝑣)))
35		f1ocnvfv2 5847	. . . . . . . . . 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 2238	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝑣))) = 𝑣)
38	37	opeq1d 3825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨(1st ‘(𝑅‘(◡𝐺‘𝑣))), (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ = ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
39	29, 38	eqtrd 2238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) = ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩)
40	39, 26	eqeltrrd 2283	. . . . 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 6252	. . . . . . 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 4883	. . . . . 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 3199	. 2 ⊢ (𝜑 → (ℤ≥‘𝐶) ⊆ dom ran 𝑅)
49	12, 48	eqssd 3210	1 ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176   ⊆ wss 3166  ⟨cop 3636   ↦ cmpt 4105  ωcom 4638   × cxp 4673  ^◡ccnv 4674  dom cdm 4675  ran crn 4676  Fun wfun 5265  ⟶wf 5267  –1-1-onto→wf1o 5270  ‘cfv 5271  (class class class)co 5944   ∈ cmpo 5946  1^st c1st 6224  2^nd c2nd 6225  freccfrec 6476  1c1 7926   + caddc 7928  ℤcz 9372  ℤ_≥cuz 9648

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649

This theorem is referenced by:  frecuzrdgdom  10563