Theorem frecuzrdgtcl 10521

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10508 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 2266	. . . . . . . . 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 10519	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
9		ffn 5410	. . . . . . . . . 10 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω)
10		fvelrnb 5611	. . . . . . . . . 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 10517	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆))
14		eleq1 2259	. . . . . . . . . 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 2614	. . . . . . . 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 3190	. . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ ((ℤ≥‘𝐶) × 𝑆))
19		xpss 4772	. . . . . 6 ⊢ ((ℤ≥‘𝐶) × 𝑆) ⊆ (V × V)
20	18, 19	sstrdi 3196	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (V × V))
21		df-rel 4671	. . . . 5 ⊢ (Rel 𝑇 ↔ 𝑇 ⊆ (V × V))
22	20, 21	sylibr 134	. . . 4 ⊢ (𝜑 → Rel 𝑇)
23	3, 4	frec2uzf1od 10515	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
24		f1ocnvdm 5831	. . . . . . . . . . 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 10517	. . . . . . . . . 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 6233	. . . . . . . . 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 2266	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅))
31		fvelrnb 5611	. . . . . . . . . . . . 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 10518	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
39	38	eqeq1d 2205	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
40		vex 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑣 ∈ V
41		vex 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
42	40, 41	opth2 4274	. . . . . . . . . . . . . . . . . . 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	biimtrdi 163	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
45		f1ocnvfv 5829	. . . . . . . . . . . . . . . . . 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 5561	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
49	48	fveq2d 5565	. . . . . . . . . . . . . . . 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 6216	. . . . . . . . . . . . . . 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 2230	. . . . . . . . . . . . 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 2614	. . . . . . . . . . 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 1888	. . . . . . . . 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 2206	. . . . . . . . . . 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 1838	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
63	62	spcegv 2852	. . . . . . . 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 1542	. . . . . . . 8 ⊢ Ⅎ𝑤⟨𝑣, 𝑧⟩ ∈ 𝑇
66	65	mo2r 2097	. . . . . . 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 4866	. . . . . . . . . . 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 5101	. . . . . . . . . 10 ⊢ dom ((ℤ≥‘𝐶) × 𝑆) ⊆ (ℤ≥‘𝐶)
71	69, 70	sstrdi 3196	. . . . . . . . 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 10520	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
77	1	eleq2d 2266	. . . . . . . . . . . . . 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 4872	. . . . . . . . . . . . 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 3190	. . . . . . . . 9 ⊢ (𝜑 → (ℤ≥‘𝐶) ⊆ dom 𝑇)
85	71, 84	eqssd 3201	. . . . . . . 8 ⊢ (𝜑 → dom 𝑇 = (ℤ≥‘𝐶))
86	85	eleq2d 2266	. . . . . . 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 4035	. . . . . . 7 ⊢ (𝑣𝑇𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑇)
89	88	mobii 2082	. . . . . 6 ⊢ (∃*𝑧 𝑣𝑇𝑧 ↔ ∃*𝑧⟨𝑣, 𝑧⟩ ∈ 𝑇)
90	67, 87, 89	3imtr4i 201	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ dom 𝑇) → ∃*𝑧 𝑣𝑇𝑧)
91	90	ralrimiva 2570	. . . 4 ⊢ (𝜑 → ∀𝑣 ∈ dom 𝑇∃*𝑧 𝑣𝑇𝑧)
92		dffun7 5286	. . . 4 ⊢ (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀𝑣 ∈ dom 𝑇∃*𝑧 𝑣𝑇𝑧))
93	22, 91, 92	sylanbrc 417	. . 3 ⊢ (𝜑 → Fun 𝑇)
94		df-fn 5262	. . 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 4897	. . . 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 5102	. . 3 ⊢ ran ((ℤ≥‘𝐶) × 𝑆) ⊆ 𝑆
99	97, 98	sstrdi 3196	. 2 ⊢ (𝜑 → ran 𝑇 ⊆ 𝑆)
100		df-f 5263	. 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 1362   = wceq 1364  ∃wex 1506  ∃*wmo 2046   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ⊆ wss 3157  ⟨cop 3626   class class class wbr 4034   ↦ cmpt 4095  ωcom 4627   × cxp 4662  ^◡ccnv 4663  dom cdm 4664  ran crn 4665  Rel wrel 4669  Fun wfun 5253   Fn wfn 5254  ⟶wf 5255  –1-1-onto→wf1o 5258  ‘cfv 5259  (class class class)co 5925   ∈ cmpo 5927  2^nd c2nd 6206  freccfrec 6457  1c1 7897   + caddc 7899  ℤcz 9343  ℤ_≥cuz 9618

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619

This theorem is referenced by:  frecuzrdg0  10522  frecuzrdgsuc  10523