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Theorem unbenlem 16969

Description: Lemma for unben 16970. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)

**Hypothesis**
Ref	Expression
unbenlem.1	⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)

**Assertion**
Ref	Expression
unbenlem	⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)

Distinct variable groups: 𝑚,𝑛,𝐴 𝑚,𝐺,𝑛 Allowed substitution hints: 𝐴(𝑥) 𝐺(𝑥)

Proof of Theorem unbenlem Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnex 12240	. . . . 5 ⊢ ℕ ∈ V
2	1	ssex 5292	. . . 4 ⊢ (𝐴 ⊆ ℕ → 𝐴 ∈ V)
3		1z 12625	. . . . . . . 8 ⊢ 1 ∈ ℤ
4		unbenlem.1	. . . . . . . 8 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)
5	3, 4	om2uzf1oi 13991	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→(ℤ_≥‘1)
6		nnuz 12902	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
7		f1oeq3 6812	. . . . . . . 8 ⊢ (ℕ = (ℤ_≥‘1) → (𝐺:ω–1-1-onto→ℕ ↔ 𝐺:ω–1-1-onto→(ℤ_≥‘1)))
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ ↔ 𝐺:ω–1-1-onto→(ℤ_≥‘1))
9	5, 8	mpbir 234	. . . . . 6 ⊢ 𝐺:ω–1-1-onto→ℕ
10		f1ocnv 6835	. . . . . 6 ⊢ (𝐺:ω–1-1-onto→ℕ → ^◡𝐺:ℕ–1-1-onto→ω)
11		f1of1 6821	. . . . . 6 ⊢ (^◡𝐺:ℕ–1-1-onto→ω → ^◡𝐺:ℕ–1-1→ω)
12	9, 10, 11	mp2b 10	. . . . 5 ⊢ ^◡𝐺:ℕ–1-1→ω
13		f1ores 6837	. . . . 5 ⊢ ((^◡𝐺:ℕ–1-1→ω ∧ 𝐴 ⊆ ℕ) → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
14	12, 13	mpan 702	. . . 4 ⊢ (𝐴 ⊆ ℕ → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
15		f1oeng 8968	. . . 4 ⊢ ((𝐴 ∈ V ∧ (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴)) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
16	2, 14, 15	syl2anc 595	. . 3 ⊢ (𝐴 ⊆ ℕ → 𝐴 ≈ (^◡𝐺 “ 𝐴))
17	16	adantr 485	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
18		imassrn 6075	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
19		dfdm4 5887	. . . . 5 ⊢ dom 𝐺 = ran ^◡𝐺
20		f1of 6822	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ → 𝐺:ω⟶ℕ)
21	9, 20	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ
22	21	fdmi 6719	. . . . 5 ⊢ dom 𝐺 = ω
23	19, 22	eqtr3i 2788	. . . 4 ⊢ ran ^◡𝐺 = ω
24	18, 23	sseqtri 3986	. . 3 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
25	3, 4	om2uzuzi 13987	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ_≥‘1))
26	25, 6	eleqtrrdi 2874	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℕ)
27		breq1 5113	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝐺‘𝑦) → (𝑚 < 𝑛 ↔ (𝐺‘𝑦) < 𝑛))
28	27	rexbidv 3189	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑦) → (∃𝑛 ∈ 𝐴 𝑚 < 𝑛 ↔ ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
29	28	rspcv 3578	. . . . . . . . . 10 ⊢ ((𝐺‘𝑦) ∈ ℕ → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
30	26, 29	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
31	30	adantr 485	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
32		f1ocnv 6835	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴) → ^◡(^◡𝐺 ↾ 𝐴):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴)
33	14, 32	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ ℕ → ^◡(^◡𝐺 ↾ 𝐴):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴)
34		f1ofun 6824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ω–1-1-onto→ℕ → Fun 𝐺)
35	9, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Fun 𝐺
36		funcnvres2 6618	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐺 → ^◡(^◡𝐺 ↾ 𝐴) = (𝐺 ↾ (^◡𝐺 “ 𝐴)))
37		f1oeq1 6810	. . . . . . . . . . . . . . . . 17 ⊢ (^◡(^◡𝐺 ↾ 𝐴) = (𝐺 ↾ (^◡𝐺 “ 𝐴)) → (^◡(^◡𝐺 ↾ 𝐴):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 ↔ (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴))
38	35, 36, 37	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ (^◡(^◡𝐺 ↾ 𝐴):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 ↔ (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴)
39	33, 38	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℕ → (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴)
40		f1ofo 6830	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–onto→𝐴)
41		forn 6797	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–onto→𝐴 → ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) = 𝐴)
42	40, 41	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) = 𝐴)
43	42	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝑛 ∈ ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) ↔ 𝑛 ∈ 𝐴))
44		f1ofn 6823	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)) Fn (^◡𝐺 “ 𝐴))
45		fvelrnb 6943	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)) Fn (^◡𝐺 “ 𝐴) → (𝑛 ∈ ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) ↔ ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛))
46	44, 45	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝑛 ∈ ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) ↔ ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛))
47	43, 46	bitr3d 284	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝑛 ∈ 𝐴 ↔ ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛))
48	39, 47	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ ℕ → (𝑛 ∈ 𝐴 ↔ ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛))
49	48	biimpa 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑛 ∈ 𝐴) → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛)
50		fvres 6902	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = (𝐺‘𝑚))
51	50	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 ↔ (𝐺‘𝑚) = 𝑛))
52	51	biimpa 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (^◡𝐺 “ 𝐴) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
53	52	adantll 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
54	24	sseli 3934	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → 𝑚 ∈ ω)
55	3, 4	om2uzlt2i 13989	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ ω) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
56	54, 55	sylan2 604	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
57		breq2 5114	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑚) = 𝑛 → ((𝐺‘𝑦) < (𝐺‘𝑚) ↔ (𝐺‘𝑦) < 𝑛))
58	56, 57	sylan9bb 518	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ (𝐺‘𝑚) = 𝑛) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < 𝑛))
59	53, 58	syldan 602	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < 𝑛))
60	59	biimparc 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺‘𝑦) < 𝑛 ∧ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛)) → 𝑦 ∈ 𝑚)
61	60	exp44 442	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑦) < 𝑛 → (𝑦 ∈ ω → (𝑚 ∈ (^◡𝐺 “ 𝐴) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → 𝑦 ∈ 𝑚))))
62	61	imp31 422	. . . . . . . . . . . . . 14 ⊢ ((((𝐺‘𝑦) < 𝑛 ∧ 𝑦 ∈ ω) ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → 𝑦 ∈ 𝑚))
63	62	reximdva 3178	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑦) < 𝑛 ∧ 𝑦 ∈ ω) → (∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
64	49, 63	syl5 35	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑦) < 𝑛 ∧ 𝑦 ∈ ω) → ((𝐴 ⊆ ℕ ∧ 𝑛 ∈ 𝐴) → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
65	64	exp4b 435	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑦) < 𝑛 → (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (𝑛 ∈ 𝐴 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))))
66	65	com4l 93	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (𝑛 ∈ 𝐴 → ((𝐺‘𝑦) < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))))
67	66	imp 411	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (𝑛 ∈ 𝐴 → ((𝐺‘𝑦) < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)))
68	67	rexlimdv 3164	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
69	31, 68	syld 48	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
70	69	ex 417	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)))
71	70	com3l 90	. . . . 5 ⊢ (𝐴 ⊆ ℕ → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → (𝑦 ∈ ω → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)))
72	71	imp 411	. . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (𝑦 ∈ ω → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
73	72	ralrimiv 3156	. . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)
74		unbnn3 9629	. . 3 ⊢ (((^◡𝐺 “ 𝐴) ⊆ ω ∧ ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚) → (^◡𝐺 “ 𝐴) ≈ ω)
75	24, 73, 74	sylancr 598	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (^◡𝐺 “ 𝐴) ≈ ω)
76		entr 9004	. 2 ⊢ ((𝐴 ≈ (^◡𝐺 “ 𝐴) ∧ (^◡𝐺 “ 𝐴) ≈ ω) → 𝐴 ≈ ω)
77	17, 75, 76	syl2anc 595	1 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ⊆ wss 3906 class class class wbr 5110 ↦ cmpt 5193 ^◡ccnv 5662 dom cdm 5663 ran crn 5664 ↾ cres 5665 “ cima 5666 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 –1-1→wf1 6535 –onto→wfo 6536 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 ωcom 7863 reccrdg 8397 ≈ cen 8941 1c1 11102 + caddc 11104 < clt 11244 ℕcn 12234 ℤ_≥cuz 12863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-n0 12506 df-z 12593 df-uz 12864

This theorem is referenced by: unben 16970

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