Theorem frecuzrdgtcl 10405

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10392 for the description of 𝐺 as the mapping from ω to (ℤ_≥‘𝐶). (Contributed by Jim Kingdon, 26-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	⊢ (𝜑 → 𝐶 ∈ ℤ)
frec2uz.2	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
frecuzrdgrrn.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
frecuzrdgrrn.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrrn.2	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgtcl.3	⊢ (𝜑 → 𝑇 = ran 𝑅)

Assertion

Ref	Expression
frecuzrdgtcl	⊢ (𝜑 → 𝑇:(ℤ≥‘𝐶)⟶𝑆)

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

Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem frecuzrdgtcl

Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecuzrdgtcl.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 = ran 𝑅)
2	1	eleq2d 2247	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝑇 ↔ 𝑧 ∈ ran 𝑅))
3		frec2uz.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℤ)
4		frec2uz.2	. . . . . . . . . . 11 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
5		frecuzrdgrrn.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
6		frecuzrdgrrn.f	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
7		frecuzrdgrrn.2	. . . . . . . . . . 11 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
8	3, 4, 5, 6, 7	frecuzrdgrcl 10403	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
9		ffn 5362	. . . . . . . . . 10 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω)
10		fvelrnb 5560	. . . . . . . . . 10 ⊢ (𝑅 Fn ω → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
11	8, 9, 10	3syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
12	2, 11	bitrd 188	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ 𝑇 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
13	3, 4, 5, 6, 7	frecuzrdgrrn 10401	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆))
14		eleq1 2240	. . . . . . . . . 10 ⊢ ((𝑅‘𝑤) = 𝑧 → ((𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)))
15	13, 14	syl5ibcom 155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)))
16	15	rexlimdva 2594	. . . . . . . 8 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)))
17	12, 16	sylbid 150	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝑇 → 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)))
18	17	ssrdv 3161	. . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ ((ℤ≥‘𝐶) × 𝑆))
19		xpss 4732	. . . . . 6 ⊢ ((ℤ≥‘𝐶) × 𝑆) ⊆ (V × V)
20	18, 19	sstrdi 3167	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (V × V))
21		df-rel 4631	. . . . 5 ⊢ (Rel 𝑇 ↔ 𝑇 ⊆ (V × V))
22	20, 21	sylibr 134	. . . 4 ⊢ (𝜑 → Rel 𝑇)
23	3, 4	frec2uzf1od 10399	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
24		f1ocnvdm 5777	. . . . . . . . . . 11 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
25	23, 24	sylan 283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
26	3, 4, 5, 6, 7	frecuzrdgrrn 10401	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (◡𝐺‘𝑣) ∈ ω) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
27	25, 26	syldan 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
28		xp2nd 6162	. . . . . . . . 9 ⊢ ((𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
29	27, 28	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
30	1	eleq2d 2247	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅))
31		fvelrnb 5560	. . . . . . . . . . . . 13 ⊢ (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
32	8, 9, 31	3syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
33	30, 32	bitrd 188	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
34	3	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝐶 ∈ ℤ)
35	5	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝐴 ∈ 𝑆)
36	6	adantlr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
37		simpr 110	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝑤 ∈ ω)
38	34, 4, 35, 36, 7, 37	frec2uzrdg 10402	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
39	38	eqeq1d 2186	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
40		vex 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑣 ∈ V
41		vex 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
42	40, 41	opth2 4238	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ ↔ ((𝐺‘𝑤) = 𝑣 ∧ (2nd ‘(𝑅‘𝑤)) = 𝑧))
43	42	simplbi 274	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
44	39, 43	syl6bi 163	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
45		f1ocnvfv 5775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
46	23, 45	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
47	44, 46	syld 45	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (◡𝐺‘𝑣) = 𝑤))
48		fveq2 5512	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
49	48	fveq2d 5516	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
50	47, 49	syl6 33	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤))))
51	50	imp 124	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
52	40, 41	op2ndd 6145	. . . . . . . . . . . . . . 15 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
53	52	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
54	51, 53	eqtr2d 2211	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
55	54	ex 115	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
56	55	rexlimdva 2594	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
57	33, 56	sylbid 150	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
58	57	alrimiv 1874	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
59	58	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
60		eqeq2 2187	. . . . . . . . . . 11 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
61	60	imbi2d 230	. . . . . . . . . 10 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
62	61	albidv 1824	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
63	62	spcegv 2825	. . . . . . . 8 ⊢ ((2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆 → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤)))
64	29, 59, 63	sylc 62	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤))
65		nfv 1528	. . . . . . . 8 ⊢ Ⅎ𝑤⟨𝑣, 𝑧⟩ ∈ 𝑇
66	65	mo2r 2078	. . . . . . 7 ⊢ (∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤) → ∃*𝑧⟨𝑣, 𝑧⟩ ∈ 𝑇)
67	64, 66	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∃*𝑧⟨𝑣, 𝑧⟩ ∈ 𝑇)
68		dmss 4823	. . . . . . . . . . 11 ⊢ (𝑇 ⊆ ((ℤ≥‘𝐶) × 𝑆) → dom 𝑇 ⊆ dom ((ℤ≥‘𝐶) × 𝑆))
69	18, 68	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑇 ⊆ dom ((ℤ≥‘𝐶) × 𝑆))
70		dmxpss 5056	. . . . . . . . . 10 ⊢ dom ((ℤ≥‘𝐶) × 𝑆) ⊆ (ℤ≥‘𝐶)
71	69, 70	sstrdi 3167	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑇 ⊆ (ℤ≥‘𝐶))
72	3	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ)
73	5	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝐴 ∈ 𝑆)
74	6	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
75		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑣 ∈ (ℤ≥‘𝐶))
76	72, 4, 73, 74, 7, 75	frecuzrdglem 10404	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
77	1	eleq2d 2247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇 ↔ ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅))
78	77	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇 ↔ ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅))
79	76, 78	mpbird 167	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇)
80		opeldmg 4829	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ V ∧ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆) → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇 → 𝑣 ∈ dom 𝑇))
81	40, 80	mpan 424	. . . . . . . . . . . 12 ⊢ ((2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆 → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇 → 𝑣 ∈ dom 𝑇))
82	29, 79, 81	sylc 62	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑣 ∈ dom 𝑇)
83	82	ex 115	. . . . . . . . . 10 ⊢ (𝜑 → (𝑣 ∈ (ℤ≥‘𝐶) → 𝑣 ∈ dom 𝑇))
84	83	ssrdv 3161	. . . . . . . . 9 ⊢ (𝜑 → (ℤ≥‘𝐶) ⊆ dom 𝑇)
85	71, 84	eqssd 3172	. . . . . . . 8 ⊢ (𝜑 → dom 𝑇 = (ℤ≥‘𝐶))
86	85	eleq2d 2247	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ dom 𝑇 ↔ 𝑣 ∈ (ℤ≥‘𝐶)))
87	86	pm5.32i 454	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ dom 𝑇) ↔ (𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)))
88		df-br 4002	. . . . . . 7 ⊢ (𝑣𝑇𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑇)
89	88	mobii 2063	. . . . . 6 ⊢ (∃*𝑧 𝑣𝑇𝑧 ↔ ∃*𝑧⟨𝑣, 𝑧⟩ ∈ 𝑇)
90	67, 87, 89	3imtr4i 201	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ dom 𝑇) → ∃*𝑧 𝑣𝑇𝑧)
91	90	ralrimiva 2550	. . . 4 ⊢ (𝜑 → ∀𝑣 ∈ dom 𝑇∃*𝑧 𝑣𝑇𝑧)
92		dffun7 5240	. . . 4 ⊢ (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀𝑣 ∈ dom 𝑇∃*𝑧 𝑣𝑇𝑧))
93	22, 91, 92	sylanbrc 417	. . 3 ⊢ (𝜑 → Fun 𝑇)
94		df-fn 5216	. . 3 ⊢ (𝑇 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑇 ∧ dom 𝑇 = (ℤ≥‘𝐶)))
95	93, 85, 94	sylanbrc 417	. 2 ⊢ (𝜑 → 𝑇 Fn (ℤ≥‘𝐶))
96		rnss 4854	. . . 4 ⊢ (𝑇 ⊆ ((ℤ≥‘𝐶) × 𝑆) → ran 𝑇 ⊆ ran ((ℤ≥‘𝐶) × 𝑆))
97	18, 96	syl 14	. . 3 ⊢ (𝜑 → ran 𝑇 ⊆ ran ((ℤ≥‘𝐶) × 𝑆))
98		rnxpss 5057	. . 3 ⊢ ran ((ℤ≥‘𝐶) × 𝑆) ⊆ 𝑆
99	97, 98	sstrdi 3167	. 2 ⊢ (𝜑 → ran 𝑇 ⊆ 𝑆)
100		df-f 5217	. 2 ⊢ (𝑇:(ℤ≥‘𝐶)⟶𝑆 ↔ (𝑇 Fn (ℤ≥‘𝐶) ∧ ran 𝑇 ⊆ 𝑆))
101	95, 99, 100	sylanbrc 417	1 ⊢ (𝜑 → 𝑇:(ℤ≥‘𝐶)⟶𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492  ∃*wmo 2027   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  Vcvv 2737   ⊆ wss 3129  ⟨cop 3595   class class class wbr 4001   ↦ cmpt 4062  ωcom 4587   × cxp 4622  ^◡ccnv 4623  dom cdm 4624  ran crn 4625  Rel wrel 4629  Fun wfun 5207   Fn wfn 5208  ⟶wf 5209  –1-1-onto→wf1o 5212  ‘cfv 5213  (class class class)co 5870   ∈ cmpo 5872  2^nd c2nd 6135  freccfrec 6386  1c1 7807   + caddc 7809  ℤcz 9247  ℤ_≥cuz 9522

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-addcom 7906  ax-addass 7908  ax-distr 7910  ax-i2m1 7911  ax-0lt1 7912  ax-0id 7914  ax-rnegex 7915  ax-cnre 7917  ax-pre-ltirr 7918  ax-pre-ltwlin 7919  ax-pre-lttrn 7920  ax-pre-ltadd 7922

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-iord 4364  df-on 4366  df-ilim 4367  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-recs 6301  df-frec 6387  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992  df-sub 8124  df-neg 8125  df-inn 8914  df-n0 9171  df-z 9248  df-uz 9523

This theorem is referenced by:  frecuzrdg0  10406  frecuzrdgsuc  10407