Theorem frecuzrdgfn 9414

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9401 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)
uzrdg.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
uzrdg.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
uzrdg.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
uzrdg.2	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgfn.3	⊢ (𝜑 → 𝑇 = ran 𝑅)

Assertion

Ref	Expression
frecuzrdgfn	⊢ (𝜑 → 𝑇 Fn (ℤ≥‘𝐶))

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)   𝑉(𝑥,𝑦)

Proof of Theorem frecuzrdgfn

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

Step	Hyp	Ref	Expression
1		frecuzrdgfn.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 = ran 𝑅)
2	1	eleq2d 2148	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ 𝑇 ↔ 𝑧 ∈ ran 𝑅))
3		frec2uz.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℤ)
4		frec2uz.2	. . . . . . . . . 10 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
5		uzrdg.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
6		uzrdg.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
7		uzrdg.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
8		uzrdg.2	. . . . . . . . . 10 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
9	3, 4, 5, 6, 7, 8	frecuzrdgrom 9412	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 Fn ω)
10		fvelrnb 5242	. . . . . . . . 9 ⊢ (𝑅 Fn ω → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
11	9, 10	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
12	2, 11	bitrd 186	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝑇 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
13	3, 4, 5, 6, 7, 8	frecuzrdgrrn 9410	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆))
14		eleq1 2141	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = 𝑧 → ((𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)))
15	13, 14	syl5ibcom 153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)))
16	15	rexlimdva 2477	. . . . . . 7 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)))
17	12, 16	sylbid 148	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 𝑇 → 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)))
18	17	ssrdv 3005	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((ℤ≥‘𝐶) × 𝑆))
19		xpss 4464	. . . . 5 ⊢ ((ℤ≥‘𝐶) × 𝑆) ⊆ (V × V)
20	18, 19	syl6ss 3011	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (V × V))
21		df-rel 4370	. . . 4 ⊢ (Rel 𝑇 ↔ 𝑇 ⊆ (V × V))
22	20, 21	sylibr 132	. . 3 ⊢ (𝜑 → Rel 𝑇)
23	3, 4	frec2uzf1od 9408	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
24		f1ocnvdm 5441	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
25	23, 24	sylan 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
26	3, 4, 5, 6, 7, 8	frecuzrdgrrn 9410	. . . . . . . . 9 ⊢ ((𝜑 ∧ (◡𝐺‘𝑣) ∈ ω) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
27	25, 26	syldan 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
28		xp2nd 5813	. . . . . . . 8 ⊢ ((𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
29	27, 28	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
30	1	eleq2d 2148	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅))
31		fvelrnb 5242	. . . . . . . . . . . 12 ⊢ (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
32	9, 31	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
33	30, 32	bitrd 186	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
34	3	adantr 270	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝐶 ∈ ℤ)
35	5	adantr 270	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝑆 ∈ 𝑉)
36	6	adantr 270	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝐴 ∈ 𝑆)
37	7	adantlr 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
38		simpr 108	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝑤 ∈ ω)
39	34, 4, 35, 36, 37, 8, 38	frec2uzrdg 9411	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
40	39	eqeq1d 2089	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
41		vex 2604	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
42		vex 2604	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ V
43	41, 42	opth2 3995	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ ↔ ((𝐺‘𝑤) = 𝑣 ∧ (2nd ‘(𝑅‘𝑤)) = 𝑧))
44	43	simplbi 268	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
45	40, 44	syl6bi 161	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
46		f1ocnvfv 5439	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
47	23, 46	sylan 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
48	45, 47	syld 44	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (◡𝐺‘𝑣) = 𝑤))
49		fveq2 5198	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
50	49	fveq2d 5202	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
51	48, 50	syl6 33	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤))))
52	51	imp 122	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
53	41, 42	op2ndd 5796	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
54	53	adantl 271	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
55	52, 54	eqtr2d 2114	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
56	55	ex 113	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
57	56	rexlimdva 2477	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
58	33, 57	sylbid 148	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
59	58	alrimiv 1795	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
60	59	adantr 270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
61		eqeq2 2090	. . . . . . . . . 10 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
62	61	imbi2d 228	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
63	62	albidv 1745	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
64	63	spcegv 2686	. . . . . . 7 ⊢ ((2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆 → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤)))
65	29, 60, 64	sylc 61	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤))
66		nfv 1461	. . . . . . 7 ⊢ Ⅎ𝑤⟨𝑣, 𝑧⟩ ∈ 𝑇
67	66	mo2r 1993	. . . . . 6 ⊢ (∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑇 → 𝑧 = 𝑤) → ∃*𝑧⟨𝑣, 𝑧⟩ ∈ 𝑇)
68	65, 67	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∃*𝑧⟨𝑣, 𝑧⟩ ∈ 𝑇)
69		dmss 4552	. . . . . . . . . 10 ⊢ (𝑇 ⊆ ((ℤ≥‘𝐶) × 𝑆) → dom 𝑇 ⊆ dom ((ℤ≥‘𝐶) × 𝑆))
70	18, 69	syl 14	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑇 ⊆ dom ((ℤ≥‘𝐶) × 𝑆))
71		dmxpss 4773	. . . . . . . . 9 ⊢ dom ((ℤ≥‘𝐶) × 𝑆) ⊆ (ℤ≥‘𝐶)
72	70, 71	syl6ss 3011	. . . . . . . 8 ⊢ (𝜑 → dom 𝑇 ⊆ (ℤ≥‘𝐶))
73	3	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ)
74	5	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑆 ∈ 𝑉)
75	6	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝐴 ∈ 𝑆)
76	7	adantlr 460	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
77		simpr 108	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑣 ∈ (ℤ≥‘𝐶))
78	73, 4, 74, 75, 76, 8, 77	frecuzrdglem 9413	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
79	1	eleq2d 2148	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇 ↔ ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅))
80	79	adantr 270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇 ↔ ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅))
81	78, 80	mpbird 165	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇)
82		opeldmg 4558	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ V ∧ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆) → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇 → 𝑣 ∈ dom 𝑇))
83	41, 82	mpan 414	. . . . . . . . . . 11 ⊢ ((2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆 → (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑇 → 𝑣 ∈ dom 𝑇))
84	29, 81, 83	sylc 61	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → 𝑣 ∈ dom 𝑇)
85	84	ex 113	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ (ℤ≥‘𝐶) → 𝑣 ∈ dom 𝑇))
86	85	ssrdv 3005	. . . . . . . 8 ⊢ (𝜑 → (ℤ≥‘𝐶) ⊆ dom 𝑇)
87	72, 86	eqssd 3016	. . . . . . 7 ⊢ (𝜑 → dom 𝑇 = (ℤ≥‘𝐶))
88	87	eleq2d 2148	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ dom 𝑇 ↔ 𝑣 ∈ (ℤ≥‘𝐶)))
89	88	pm5.32i 441	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ dom 𝑇) ↔ (𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)))
90		df-br 3786	. . . . . 6 ⊢ (𝑣𝑇𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ 𝑇)
91	90	mobii 1978	. . . . 5 ⊢ (∃*𝑧 𝑣𝑇𝑧 ↔ ∃*𝑧⟨𝑣, 𝑧⟩ ∈ 𝑇)
92	68, 89, 91	3imtr4i 199	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ dom 𝑇) → ∃*𝑧 𝑣𝑇𝑧)
93	92	ralrimiva 2434	. . 3 ⊢ (𝜑 → ∀𝑣 ∈ dom 𝑇∃*𝑧 𝑣𝑇𝑧)
94		dffun7 4948	. . 3 ⊢ (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀𝑣 ∈ dom 𝑇∃*𝑧 𝑣𝑇𝑧))
95	22, 93, 94	sylanbrc 408	. 2 ⊢ (𝜑 → Fun 𝑇)
96		df-fn 4925	. 2 ⊢ (𝑇 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑇 ∧ dom 𝑇 = (ℤ≥‘𝐶)))
97	95, 87, 96	sylanbrc 408	1 ⊢ (𝜑 → 𝑇 Fn (ℤ≥‘𝐶))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃*wmo 1942  ∀wral 2348  ∃wrex 2349  Vcvv 2601   ⊆ wss 2973  ⟨cop 3401   class class class wbr 3785   ↦ cmpt 3839  ωcom 4331   × cxp 4361  ^◡ccnv 4362  dom cdm 4363  ran crn 4364  Rel wrel 4368  Fun wfun 4916   Fn wfn 4917  –1-1-onto→wf1o 4921  ‘cfv 4922  (class class class)co 5532   ↦ cmpt2 5534  2^nd c2nd 5786  freccfrec 6000  1c1 6982   + caddc 6984  ℤcz 8351  ℤ_≥cuz 8619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620

This theorem is referenced by:  frecuzrdgcl  9415  frecuzrdg0  9416  frecuzrdgsuc  9417  iseqfn  9441