Theorem frecuzrdgfn 8859

Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 8846 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 2104	. . . . . . . 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 8857	. . . . . . . . 9 ⊢ (φ → 𝑅 Fn 𝜔)
10		fvelrnb 5164	. . . . . . . . 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 177	. . . . . . 7 ⊢ (φ → (z ∈ 𝑇 ↔ ∃w ∈ 𝜔 (𝑅‘w) = z))
13	3, 4, 5, 6, 7, 8	frecuzrdgrrn 8855	. . . . . . . . 9 ⊢ ((φ ∧ w ∈ 𝜔) → (𝑅‘w) ∈ ((ℤ≥‘𝐶) × 𝑆))
14		eleq1 2097	. . . . . . . . 9 ⊢ ((𝑅‘w) = z → ((𝑅‘w) ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ z ∈ ((ℤ≥‘𝐶) × 𝑆)))
15	13, 14	syl5ibcom 144	. . . . . . . 8 ⊢ ((φ ∧ w ∈ 𝜔) → ((𝑅‘w) = z → z ∈ ((ℤ≥‘𝐶) × 𝑆)))
16	15	rexlimdva 2427	. . . . . . 7 ⊢ (φ → (∃w ∈ 𝜔 (𝑅‘w) = z → z ∈ ((ℤ≥‘𝐶) × 𝑆)))
17	12, 16	sylbid 139	. . . . . 6 ⊢ (φ → (z ∈ 𝑇 → z ∈ ((ℤ≥‘𝐶) × 𝑆)))
18	17	ssrdv 2945	. . . . 5 ⊢ (φ → 𝑇 ⊆ ((ℤ≥‘𝐶) × 𝑆))
19		xpss 4389	. . . . 5 ⊢ ((ℤ≥‘𝐶) × 𝑆) ⊆ (V × V)
20	18, 19	syl6ss 2951	. . . 4 ⊢ (φ → 𝑇 ⊆ (V × V))
21		df-rel 4295	. . . 4 ⊢ (Rel 𝑇 ↔ 𝑇 ⊆ (V × V))
22	20, 21	sylibr 137	. . 3 ⊢ (φ → Rel 𝑇)
23	3, 4	frec2uzf1od 8853	. . . . . . . . . 10 ⊢ (φ → 𝐺:𝜔–1-1-onto→(ℤ≥‘𝐶))
24		f1ocnvdm 5364	. . . . . . . . . 10 ⊢ ((𝐺:𝜔–1-1-onto→(ℤ≥‘𝐶) ∧ v ∈ (ℤ≥‘𝐶)) → (◡𝐺‘v) ∈ 𝜔)
25	23, 24	sylan 267	. . . . . . . . 9 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → (◡𝐺‘v) ∈ 𝜔)
26	3, 4, 5, 6, 7, 8	frecuzrdgrrn 8855	. . . . . . . . 9 ⊢ ((φ ∧ (◡𝐺‘v) ∈ 𝜔) → (𝑅‘(◡𝐺‘v)) ∈ ((ℤ≥‘𝐶) × 𝑆))
27	25, 26	syldan 266	. . . . . . . 8 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘v)) ∈ ((ℤ≥‘𝐶) × 𝑆))
28		xp2nd 5735	. . . . . . . 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 2104	. . . . . . . . . . 11 ⊢ (φ → (⟨v, z⟩ ∈ 𝑇 ↔ ⟨v, z⟩ ∈ ran 𝑅))
31		fvelrnb 5164	. . . . . . . . . . . 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 177	. . . . . . . . . 10 ⊢ (φ → (⟨v, z⟩ ∈ 𝑇 ↔ ∃w ∈ 𝜔 (𝑅‘w) = ⟨v, z⟩))
34	3	adantr 261	. . . . . . . . . . . . . . . . . . 19 ⊢ ((φ ∧ w ∈ 𝜔) → 𝐶 ∈ ℤ)
35	5	adantr 261	. . . . . . . . . . . . . . . . . . 19 ⊢ ((φ ∧ w ∈ 𝜔) → 𝑆 ∈ 𝑉)
36	6	adantr 261	. . . . . . . . . . . . . . . . . . 19 ⊢ ((φ ∧ w ∈ 𝜔) → A ∈ 𝑆)
37	7	adantlr 446	. . . . . . . . . . . . . . . . . . 19 ⊢ (((φ ∧ w ∈ 𝜔) ∧ (x ∈ (ℤ≥‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆)
38		simpr 103	. . . . . . . . . . . . . . . . . . 19 ⊢ ((φ ∧ w ∈ 𝜔) → w ∈ 𝜔)
39	34, 4, 35, 36, 37, 8, 38	frec2uzrdg 8856	. . . . . . . . . . . . . . . . . 18 ⊢ ((φ ∧ w ∈ 𝜔) → (𝑅‘w) = ⟨(𝐺‘w), (2nd ‘(𝑅‘w))⟩)
40	39	eqeq1d 2045	. . . . . . . . . . . . . . . . 17 ⊢ ((φ ∧ w ∈ 𝜔) → ((𝑅‘w) = ⟨v, z⟩ ↔ ⟨(𝐺‘w), (2nd ‘(𝑅‘w))⟩ = ⟨v, z⟩))
41		vex 2554	. . . . . . . . . . . . . . . . . . 19 ⊢ v ∈ V
42		vex 2554	. . . . . . . . . . . . . . . . . . 19 ⊢ z ∈ V
43	41, 42	opth2 3968	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨(𝐺‘w), (2nd ‘(𝑅‘w))⟩ = ⟨v, z⟩ ↔ ((𝐺‘w) = v ∧ (2nd ‘(𝑅‘w)) = z))
44	43	simplbi 259	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(𝐺‘w), (2nd ‘(𝑅‘w))⟩ = ⟨v, z⟩ → (𝐺‘w) = v)
45	40, 44	syl6bi 152	. . . . . . . . . . . . . . . 16 ⊢ ((φ ∧ w ∈ 𝜔) → ((𝑅‘w) = ⟨v, z⟩ → (𝐺‘w) = v))
46		f1ocnvfv 5362	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺:𝜔–1-1-onto→(ℤ≥‘𝐶) ∧ w ∈ 𝜔) → ((𝐺‘w) = v → (◡𝐺‘v) = w))
47	23, 46	sylan 267	. . . . . . . . . . . . . . . 16 ⊢ ((φ ∧ w ∈ 𝜔) → ((𝐺‘w) = v → (◡𝐺‘v) = w))
48	45, 47	syld 40	. . . . . . . . . . . . . . 15 ⊢ ((φ ∧ w ∈ 𝜔) → ((𝑅‘w) = ⟨v, z⟩ → (◡𝐺‘v) = w))
49		fveq2 5121	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝐺‘v) = w → (𝑅‘(◡𝐺‘v)) = (𝑅‘w))
50	49	fveq2d 5125	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘v) = w → (2nd ‘(𝑅‘(◡𝐺‘v))) = (2nd ‘(𝑅‘w)))
51	48, 50	syl6 29	. . . . . . . . . . . . . 14 ⊢ ((φ ∧ w ∈ 𝜔) → ((𝑅‘w) = ⟨v, z⟩ → (2nd ‘(𝑅‘(◡𝐺‘v))) = (2nd ‘(𝑅‘w))))
52	51	imp 115	. . . . . . . . . . . . 13 ⊢ (((φ ∧ w ∈ 𝜔) ∧ (𝑅‘w) = ⟨v, z⟩) → (2nd ‘(𝑅‘(◡𝐺‘v))) = (2nd ‘(𝑅‘w)))
53	41, 42	op2ndd 5718	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘w) = ⟨v, z⟩ → (2nd ‘(𝑅‘w)) = z)
54	53	adantl 262	. . . . . . . . . . . . 13 ⊢ (((φ ∧ w ∈ 𝜔) ∧ (𝑅‘w) = ⟨v, z⟩) → (2nd ‘(𝑅‘w)) = z)
55	52, 54	eqtr2d 2070	. . . . . . . . . . . 12 ⊢ (((φ ∧ w ∈ 𝜔) ∧ (𝑅‘w) = ⟨v, z⟩) → z = (2nd ‘(𝑅‘(◡𝐺‘v))))
56	55	ex 108	. . . . . . . . . . 11 ⊢ ((φ ∧ w ∈ 𝜔) → ((𝑅‘w) = ⟨v, z⟩ → z = (2nd ‘(𝑅‘(◡𝐺‘v)))))
57	56	rexlimdva 2427	. . . . . . . . . 10 ⊢ (φ → (∃w ∈ 𝜔 (𝑅‘w) = ⟨v, z⟩ → z = (2nd ‘(𝑅‘(◡𝐺‘v)))))
58	33, 57	sylbid 139	. . . . . . . . 9 ⊢ (φ → (⟨v, z⟩ ∈ 𝑇 → z = (2nd ‘(𝑅‘(◡𝐺‘v)))))
59	58	alrimiv 1751	. . . . . . . 8 ⊢ (φ → ∀z(⟨v, z⟩ ∈ 𝑇 → z = (2nd ‘(𝑅‘(◡𝐺‘v)))))
60	59	adantr 261	. . . . . . 7 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → ∀z(⟨v, z⟩ ∈ 𝑇 → z = (2nd ‘(𝑅‘(◡𝐺‘v)))))
61		eqeq2 2046	. . . . . . . . . 10 ⊢ (w = (2nd ‘(𝑅‘(◡𝐺‘v))) → (z = w ↔ z = (2nd ‘(𝑅‘(◡𝐺‘v)))))
62	61	imbi2d 219	. . . . . . . . 9 ⊢ (w = (2nd ‘(𝑅‘(◡𝐺‘v))) → ((⟨v, z⟩ ∈ 𝑇 → z = w) ↔ (⟨v, z⟩ ∈ 𝑇 → z = (2nd ‘(𝑅‘(◡𝐺‘v))))))
63	62	albidv 1702	. . . . . . . 8 ⊢ (w = (2nd ‘(𝑅‘(◡𝐺‘v))) → (∀z(⟨v, z⟩ ∈ 𝑇 → z = w) ↔ ∀z(⟨v, z⟩ ∈ 𝑇 → z = (2nd ‘(𝑅‘(◡𝐺‘v))))))
64	63	spcegv 2635	. . . . . . 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 56	. . . . . 6 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → ∃w∀z(⟨v, z⟩ ∈ 𝑇 → z = w))
66		nfv 1418	. . . . . . 7 ⊢ Ⅎw⟨v, z⟩ ∈ 𝑇
67	66	mo2r 1949	. . . . . 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 4477	. . . . . . . . . 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 4696	. . . . . . . . 9 ⊢ dom ((ℤ≥‘𝐶) × 𝑆) ⊆ (ℤ≥‘𝐶)
72	70, 71	syl6ss 2951	. . . . . . . 8 ⊢ (φ → dom 𝑇 ⊆ (ℤ≥‘𝐶))
73	3	adantr 261	. . . . . . . . . . . . 13 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ)
74	5	adantr 261	. . . . . . . . . . . . 13 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → 𝑆 ∈ 𝑉)
75	6	adantr 261	. . . . . . . . . . . . 13 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → A ∈ 𝑆)
76	7	adantlr 446	. . . . . . . . . . . . 13 ⊢ (((φ ∧ v ∈ (ℤ≥‘𝐶)) ∧ (x ∈ (ℤ≥‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆)
77		simpr 103	. . . . . . . . . . . . 13 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → v ∈ (ℤ≥‘𝐶))
78	73, 4, 74, 75, 76, 8, 77	frecuzrdglem 8858	. . . . . . . . . . . 12 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → ⟨v, (2nd ‘(𝑅‘(◡𝐺‘v)))⟩ ∈ ran 𝑅)
79	1	eleq2d 2104	. . . . . . . . . . . . 13 ⊢ (φ → (⟨v, (2nd ‘(𝑅‘(◡𝐺‘v)))⟩ ∈ 𝑇 ↔ ⟨v, (2nd ‘(𝑅‘(◡𝐺‘v)))⟩ ∈ ran 𝑅))
80	79	adantr 261	. . . . . . . . . . . 12 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → (⟨v, (2nd ‘(𝑅‘(◡𝐺‘v)))⟩ ∈ 𝑇 ↔ ⟨v, (2nd ‘(𝑅‘(◡𝐺‘v)))⟩ ∈ ran 𝑅))
81	78, 80	mpbird 156	. . . . . . . . . . 11 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → ⟨v, (2nd ‘(𝑅‘(◡𝐺‘v)))⟩ ∈ 𝑇)
82		opeldmg 4483	. . . . . . . . . . . 12 ⊢ ((v ∈ V ∧ (2nd ‘(𝑅‘(◡𝐺‘v))) ∈ 𝑆) → (⟨v, (2nd ‘(𝑅‘(◡𝐺‘v)))⟩ ∈ 𝑇 → v ∈ dom 𝑇))
83	41, 82	mpan 400	. . . . . . . . . . 11 ⊢ ((2nd ‘(𝑅‘(◡𝐺‘v))) ∈ 𝑆 → (⟨v, (2nd ‘(𝑅‘(◡𝐺‘v)))⟩ ∈ 𝑇 → v ∈ dom 𝑇))
84	29, 81, 83	sylc 56	. . . . . . . . . 10 ⊢ ((φ ∧ v ∈ (ℤ≥‘𝐶)) → v ∈ dom 𝑇)
85	84	ex 108	. . . . . . . . 9 ⊢ (φ → (v ∈ (ℤ≥‘𝐶) → v ∈ dom 𝑇))
86	85	ssrdv 2945	. . . . . . . 8 ⊢ (φ → (ℤ≥‘𝐶) ⊆ dom 𝑇)
87	72, 86	eqssd 2956	. . . . . . 7 ⊢ (φ → dom 𝑇 = (ℤ≥‘𝐶))
88	87	eleq2d 2104	. . . . . 6 ⊢ (φ → (v ∈ dom 𝑇 ↔ v ∈ (ℤ≥‘𝐶)))
89	88	pm5.32i 427	. . . . 5 ⊢ ((φ ∧ v ∈ dom 𝑇) ↔ (φ ∧ v ∈ (ℤ≥‘𝐶)))
90		df-br 3756	. . . . . 6 ⊢ (v𝑇z ↔ ⟨v, z⟩ ∈ 𝑇)
91	90	mobii 1934	. . . . 5 ⊢ (∃*z v𝑇z ↔ ∃*z⟨v, z⟩ ∈ 𝑇)
92	68, 89, 91	3imtr4i 190	. . . 4 ⊢ ((φ ∧ v ∈ dom 𝑇) → ∃*z v𝑇z)
93	92	ralrimiva 2386	. . 3 ⊢ (φ → ∀v ∈ dom 𝑇∃*z v𝑇z)
94		dffun7 4871	. . 3 ⊢ (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀v ∈ dom 𝑇∃*z v𝑇z))
95	22, 93, 94	sylanbrc 394	. 2 ⊢ (φ → Fun 𝑇)
96		df-fn 4848	. 2 ⊢ (𝑇 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑇 ∧ dom 𝑇 = (ℤ≥‘𝐶)))
97	95, 87, 96	sylanbrc 394	1 ⊢ (φ → 𝑇 Fn (ℤ≥‘𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  ∀wral 2300  ∃wrex 2301  Vcvv 2551   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755   ↦ cmpt 3809  𝜔com 4256   × cxp 4286  ^◡ccnv 4287  dom cdm 4288  ran crn 4289  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840  –1-1-onto→wf1o 4844  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  2^nd c2nd 5708  freccfrec 5917  1c1 6692   + caddc 6694  ℤcz 8001  ℤ_≥cuz 8229

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6754  ax-resscn 6755  ax-1cn 6756  ax-1re 6757  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-addcom 6763  ax-addass 6765  ax-distr 6767  ax-i2m1 6768  ax-0id 6771  ax-rnegex 6772  ax-cnre 6774  ax-pre-ltirr 6775  ax-pre-ltwlin 6776  ax-pre-lttrn 6777  ax-pre-ltadd 6779

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-iltp 6452  df-enr 6634  df-nr 6635  df-ltr 6638  df-0r 6639  df-1r 6640  df-0 6698  df-1 6699  df-r 6701  df-lt 6704  df-pnf 6839  df-mnf 6840  df-xr 6841  df-ltxr 6842  df-le 6843  df-sub 6961  df-neg 6962  df-inn 7676  df-n0 7938  df-z 8002  df-uz 8230

This theorem is referenced by:  frecuzrdgcl  8860  frecuzrdg0  8861  frecuzrdgsuc  8862  iseqfn  8881