Theorem frecuzrdgfunlem 10781

Description: The recursive definition generator on upper integers produces a a function. (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⟩)
frecuzrdgfunlem.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)

Assertion

Ref	Expression
frecuzrdgfunlem	⊢ (𝜑 → Fun ran 𝑅)

Distinct variable groups:   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺,𝑦   𝑥,𝑅,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem frecuzrdgfunlem

Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and 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 10777	. . . . 5 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
7		frn 5517	. . . . 5 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
8	6, 7	syl 14	. . . 4 ⊢ (𝜑 → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
9		xpss 4858	. . . 4 ⊢ ((ℤ≥‘𝐶) × 𝑆) ⊆ (V × V)
10	8, 9	sstrdi 3250	. . 3 ⊢ (𝜑 → ran 𝑅 ⊆ (V × V))
11		df-rel 4756	. . 3 ⊢ (Rel ran 𝑅 ↔ ran 𝑅 ⊆ (V × V))
12	10, 11	sylibr 134	. 2 ⊢ (𝜑 → Rel ran 𝑅)
13		frecuzrdgfunlem.g	. . . . . . . . . 10 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
14	1, 13	frec2uzf1od 10768	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
15		f1ocnvdm 5954	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
16	14, 15	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
17	6	ffvelcdmda 5812	. . . . . . . 8 ⊢ ((𝜑 ∧ (◡𝐺‘𝑣) ∈ ω) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
18	16, 17	syldan 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
19		xp2nd 6360	. . . . . . 7 ⊢ ((𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
20	18, 19	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
21		ffn 5508	. . . . . . . . . 10 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω)
22		fvelrnb 5724	. . . . . . . . . 10 ⊢ (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
23	6, 21, 22	3syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
24	6	ffvelcdmda 5812	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆))
25		1st2nd2 6369	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆) → (𝑅‘𝑤) = ⟨(1st ‘(𝑅‘𝑤)), (2nd ‘(𝑅‘𝑤))⟩)
26	24, 25	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(1st ‘(𝑅‘𝑤)), (2nd ‘(𝑅‘𝑤))⟩)
27	1	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝐶 ∈ ℤ)
28	2	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝐴 ∈ 𝑆)
29	3	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝑆 ⊆ 𝑇)
30	4	adantlr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
31		simpr 110	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝑤 ∈ ω)
32	27, 28, 29, 30, 5, 31, 13	frecuzrdgg 10778	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤))
33	32	opeq1d 3889	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ⟨(1st ‘(𝑅‘𝑤)), (2nd ‘(𝑅‘𝑤))⟩ = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
34	26, 33	eqtrd 2265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
35	34	eqeq1d 2241	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
36		vex 2816	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑣 ∈ V
37		vex 2816	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
38	36, 37	opth2 4356	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ ↔ ((𝐺‘𝑤) = 𝑣 ∧ (2nd ‘(𝑅‘𝑤)) = 𝑧))
39	38	simplbi 274	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
40	35, 39	biimtrdi 163	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
41		f1ocnvfv 5952	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
42	14, 41	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
43	40, 42	syld 45	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (◡𝐺‘𝑣) = 𝑤))
44		fveq2 5670	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
45	44	fveq2d 5674	. . . . . . . . . . . . . 14 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
46	43, 45	syl6 33	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤))))
47	46	imp 124	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
48	36, 37	op2ndd 6343	. . . . . . . . . . . . 13 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
49	48	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
50	47, 49	eqtr2d 2266	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
51	50	ex 115	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
52	51	rexlimdva 2660	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
53	23, 52	sylbid 150	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
54	53	alrimiv 1923	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
55	54	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
56		eqeq2 2242	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
57	56	imbi2d 230	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
58	57	albidv 1873	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
59	58	spcegv 2905	. . . . . 6 ⊢ ((2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆 → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤)))
60	20, 55, 59	sylc 62	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤))
61		nfv 1577	. . . . . 6 ⊢ Ⅎ𝑤⟨𝑣, 𝑧⟩ ∈ ran 𝑅
62	61	mo2r 2133	. . . . 5 ⊢ (∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤) → ∃*𝑧⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
63	60, 62	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∃*𝑧⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
64	1, 2, 3, 4, 5	frecuzrdgdom 10780	. . . . . 6 ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))
65	64	eleq2d 2302	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ dom ran 𝑅 ↔ 𝑣 ∈ (ℤ≥‘𝐶)))
66	65	pm5.32i 454	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ dom ran 𝑅) ↔ (𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)))
67		df-br 4110	. . . . 5 ⊢ (𝑣ran 𝑅 𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
68	67	mobii 2117	. . . 4 ⊢ (∃*𝑧 𝑣ran 𝑅 𝑧 ↔ ∃*𝑧⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
69	63, 66, 68	3imtr4i 201	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ dom ran 𝑅) → ∃*𝑧 𝑣ran 𝑅 𝑧)
70	69	ralrimiva 2615	. 2 ⊢ (𝜑 → ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧)
71		dffun7 5379	. 2 ⊢ (Fun ran 𝑅 ↔ (Rel ran 𝑅 ∧ ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧))
72	12, 70, 71	sylanbrc 417	1 ⊢ (𝜑 → Fun ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃wex 1541  ∃*wmo 2081   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  Vcvv 2813   ⊆ wss 3211  ⟨cop 3692   class class class wbr 4109   ↦ cmpt 4171  ωcom 4712   × cxp 4747  ^◡ccnv 4748  dom cdm 4749  ran crn 4750  Rel wrel 4754  Fun wfun 5346   Fn wfn 5347  ⟶wf 5348  –1-1-onto→wf1o 5351  ‘cfv 5352  (class class class)co 6050   ∈ cmpo 6052  1^st c1st 6332  2^nd c2nd 6333  freccfrec 6621  1c1 8128   + caddc 8130  ℤcz 9577  ℤ_≥cuz 9853

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854

This theorem is referenced by:  frecuzrdgfun  10782