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Theorem unbenlem 16879

Description: Lemma for unben 16880. (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 12180	. . . . 5 ⊢ ℕ ∈ V
2	1	ssex 5262	. . . 4 ⊢ (𝐴 ⊆ ℕ → 𝐴 ∈ V)
3		1z 12557	. . . . . . . 8 ⊢ 1 ∈ ℤ
4		unbenlem.1	. . . . . . . 8 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)
5	3, 4	om2uzf1oi 13915	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→(ℤ_≥‘1)
6		nnuz 12827	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
7		f1oeq3 6770	. . . . . . . 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 231	. . . . . 6 ⊢ 𝐺:ω–1-1-onto→ℕ
10		f1ocnv 6792	. . . . . 6 ⊢ (𝐺:ω–1-1-onto→ℕ → ^◡𝐺:ℕ–1-1-onto→ω)
11		f1of1 6779	. . . . . 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 6794	. . . . 5 ⊢ ((^◡𝐺:ℕ–1-1→ω ∧ 𝐴 ⊆ ℕ) → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
14	12, 13	mpan 691	. . . 4 ⊢ (𝐴 ⊆ ℕ → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
15		f1oeng 8917	. . . 4 ⊢ ((𝐴 ∈ V ∧ (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴)) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
16	2, 14, 15	syl2anc 585	. . 3 ⊢ (𝐴 ⊆ ℕ → 𝐴 ≈ (^◡𝐺 “ 𝐴))
17	16	adantr 480	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
18		imassrn 6036	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
19		dfdm4 5850	. . . . 5 ⊢ dom 𝐺 = ran ^◡𝐺
20		f1of 6780	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ → 𝐺:ω⟶ℕ)
21	9, 20	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ
22	21	fdmi 6679	. . . . 5 ⊢ dom 𝐺 = ω
23	19, 22	eqtr3i 2761	. . . 4 ⊢ ran ^◡𝐺 = ω
24	18, 23	sseqtri 3970	. . 3 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
25	3, 4	om2uzuzi 13911	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ_≥‘1))
26	25, 6	eleqtrrdi 2847	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℕ)
27		breq1 5088	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝐺‘𝑦) → (𝑚 < 𝑛 ↔ (𝐺‘𝑦) < 𝑛))
28	27	rexbidv 3161	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑦) → (∃𝑛 ∈ 𝐴 𝑚 < 𝑛 ↔ ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
29	28	rspcv 3560	. . . . . . . . . 10 ⊢ ((𝐺‘𝑦) ∈ ℕ → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
30	26, 29	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
32		f1ocnv 6792	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴) → ^◡(^◡𝐺 ↾ 𝐴):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴)
33	14, 32	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ ℕ → ^◡(^◡𝐺 ↾ 𝐴):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴)
34		f1ofun 6782	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ω–1-1-onto→ℕ → Fun 𝐺)
35	9, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Fun 𝐺
36		funcnvres2 6578	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐺 → ^◡(^◡𝐺 ↾ 𝐴) = (𝐺 ↾ (^◡𝐺 “ 𝐴)))
37		f1oeq1 6768	. . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℕ → (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴)
40		f1ofo 6787	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–onto→𝐴)
41		forn 6755	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–onto→𝐴 → ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) = 𝐴)
42	40, 41	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) = 𝐴)
43	42	eleq2d 2822	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝑛 ∈ ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) ↔ 𝑛 ∈ 𝐴))
44		f1ofn 6781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)) Fn (^◡𝐺 “ 𝐴))
45		fvelrnb 6900	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)) Fn (^◡𝐺 “ 𝐴) → (𝑛 ∈ ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) ↔ ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛))
46	44, 45	syl 17	. . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝑛 ∈ 𝐴 ↔ ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛))
48	39, 47	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ ℕ → (𝑛 ∈ 𝐴 ↔ ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛))
49	48	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑛 ∈ 𝐴) → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛)
50		fvres 6859	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = (𝐺‘𝑚))
51	50	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 ↔ (𝐺‘𝑚) = 𝑛))
52	51	biimpa 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (^◡𝐺 “ 𝐴) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
53	52	adantll 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
54	24	sseli 3917	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → 𝑚 ∈ ω)
55	3, 4	om2uzlt2i 13913	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ ω) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
56	54, 55	sylan2 594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
57		breq2 5089	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑚) = 𝑛 → ((𝐺‘𝑦) < (𝐺‘𝑚) ↔ (𝐺‘𝑦) < 𝑛))
58	56, 57	sylan9bb 509	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ (𝐺‘𝑚) = 𝑛) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < 𝑛))
59	53, 58	syldan 592	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < 𝑛))
60	59	biimparc 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺‘𝑦) < 𝑛 ∧ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛)) → 𝑦 ∈ 𝑚)
61	60	exp44 437	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑦) < 𝑛 → (𝑦 ∈ ω → (𝑚 ∈ (^◡𝐺 “ 𝐴) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → 𝑦 ∈ 𝑚))))
62	61	imp31 417	. . . . . . . . . . . . . 14 ⊢ ((((𝐺‘𝑦) < 𝑛 ∧ 𝑦 ∈ ω) ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → 𝑦 ∈ 𝑚))
63	62	reximdva 3150	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑦) < 𝑛 ∧ 𝑦 ∈ ω) → (∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
64	49, 63	syl5 34	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑦) < 𝑛 ∧ 𝑦 ∈ ω) → ((𝐴 ⊆ ℕ ∧ 𝑛 ∈ 𝐴) → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
65	64	exp4b 430	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑦) < 𝑛 → (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (𝑛 ∈ 𝐴 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))))
66	65	com4l 92	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (𝑛 ∈ 𝐴 → ((𝐺‘𝑦) < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))))
67	66	imp 406	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (𝑛 ∈ 𝐴 → ((𝐺‘𝑦) < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)))
68	67	rexlimdv 3136	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
69	31, 68	syld 47	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
70	69	ex 412	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)))
71	70	com3l 89	. . . . 5 ⊢ (𝐴 ⊆ ℕ → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → (𝑦 ∈ ω → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)))
72	71	imp 406	. . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (𝑦 ∈ ω → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
73	72	ralrimiv 3128	. . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)
74		unbnn3 9580	. . 3 ⊢ (((^◡𝐺 “ 𝐴) ⊆ ω ∧ ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚) → (^◡𝐺 “ 𝐴) ≈ ω)
75	24, 73, 74	sylancr 588	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (^◡𝐺 “ 𝐴) ≈ ω)
76		entr 8953	. 2 ⊢ ((𝐴 ≈ (^◡𝐺 “ 𝐴) ∧ (^◡𝐺 “ 𝐴) ≈ ω) → 𝐴 ≈ ω)
77	17, 75, 76	syl2anc 585	1 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ⊆ wss 3889 class class class wbr 5085 ↦ cmpt 5166 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 “ cima 5634 Fun wfun 6492 Fn wfn 6493 ⟶wf 6494 –1-1→wf1 6495 –onto→wfo 6496 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 ωcom 7817 reccrdg 8348 ≈ cen 8890 1c1 11039 + caddc 11041 < clt 11179 ℕcn 12174 ℤ_≥cuz 12788

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789

This theorem is referenced by: unben 16880

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