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Theorem unbenlem 16824

Description: Lemma for unben 16825. (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 12140	. . . . 5 ⊢ ℕ ∈ V
2	1	ssex 5263	. . . 4 ⊢ (𝐴 ⊆ ℕ → 𝐴 ∈ V)
3		1z 12510	. . . . . . . 8 ⊢ 1 ∈ ℤ
4		unbenlem.1	. . . . . . . 8 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)
5	3, 4	om2uzf1oi 13864	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→(ℤ_≥‘1)
6		nnuz 12779	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
7		f1oeq3 6760	. . . . . . . 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 6782	. . . . . 6 ⊢ (𝐺:ω–1-1-onto→ℕ → ^◡𝐺:ℕ–1-1-onto→ω)
11		f1of1 6769	. . . . . 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 6784	. . . . 5 ⊢ ((^◡𝐺:ℕ–1-1→ω ∧ 𝐴 ⊆ ℕ) → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
14	12, 13	mpan 690	. . . 4 ⊢ (𝐴 ⊆ ℕ → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
15		f1oeng 8901	. . . 4 ⊢ ((𝐴 ∈ V ∧ (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴)) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
16	2, 14, 15	syl2anc 584	. . 3 ⊢ (𝐴 ⊆ ℕ → 𝐴 ≈ (^◡𝐺 “ 𝐴))
17	16	adantr 480	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
18		imassrn 6026	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
19		dfdm4 5841	. . . . 5 ⊢ dom 𝐺 = ran ^◡𝐺
20		f1of 6770	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ → 𝐺:ω⟶ℕ)
21	9, 20	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ
22	21	fdmi 6669	. . . . 5 ⊢ dom 𝐺 = ω
23	19, 22	eqtr3i 2758	. . . 4 ⊢ ran ^◡𝐺 = ω
24	18, 23	sseqtri 3979	. . 3 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
25	3, 4	om2uzuzi 13860	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ_≥‘1))
26	25, 6	eleqtrrdi 2844	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℕ)
27		breq1 5098	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝐺‘𝑦) → (𝑚 < 𝑛 ↔ (𝐺‘𝑦) < 𝑛))
28	27	rexbidv 3157	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑦) → (∃𝑛 ∈ 𝐴 𝑚 < 𝑛 ↔ ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
29	28	rspcv 3569	. . . . . . . . . 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 6782	. . . . . . . . . . . . . . . . 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 6772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ω–1-1-onto→ℕ → Fun 𝐺)
35	9, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Fun 𝐺
36		funcnvres2 6568	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐺 → ^◡(^◡𝐺 ↾ 𝐴) = (𝐺 ↾ (^◡𝐺 “ 𝐴)))
37		f1oeq1 6758	. . . . . . . . . . . . . . . . 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 6777	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–onto→𝐴)
41		forn 6745	. . . . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝑛 ∈ ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) ↔ 𝑛 ∈ 𝐴))
44		f1ofn 6771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)) Fn (^◡𝐺 “ 𝐴))
45		fvelrnb 6890	. . . . . . . . . . . . . . . . 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 6849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = (𝐺‘𝑚))
51	50	eqeq1d 2735	. . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
54	24	sseli 3926	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → 𝑚 ∈ ω)
55	3, 4	om2uzlt2i 13862	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ ω) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
56	54, 55	sylan2 593	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
57		breq2 5099	. . . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . 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 3132	. . . . . . . 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 3124	. . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)
74		unbnn3 9558	. . 3 ⊢ (((^◡𝐺 “ 𝐴) ⊆ ω ∧ ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚) → (^◡𝐺 “ 𝐴) ≈ ω)
75	24, 73, 74	sylancr 587	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (^◡𝐺 “ 𝐴) ≈ ω)
76		entr 8937	. 2 ⊢ ((𝐴 ≈ (^◡𝐺 “ 𝐴) ∧ (^◡𝐺 “ 𝐴) ≈ ω) → 𝐴 ≈ ω)
77	17, 75, 76	syl2anc 584	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 1541 ∈ wcel 2113 ∀wral 3048 ∃wrex 3057 Vcvv 3437 ⊆ wss 3898 class class class wbr 5095 ↦ cmpt 5176 ^◡ccnv 5620 dom cdm 5621 ran crn 5622 ↾ cres 5623 “ cima 5624 Fun wfun 6482 Fn wfn 6483 ⟶wf 6484 –1-1→wf1 6485 –onto→wfo 6486 –1-1-onto→wf1o 6487 ‘cfv 6488 (class class class)co 7354 ωcom 7804 reccrdg 8336 ≈ cen 8874 1c1 11016 + caddc 11018 < clt 11155 ℕcn 12134 ℤ_≥cuz 12740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-inf2 9540 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7805 df-2nd 7930 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-er 8630 df-en 8878 df-dom 8879 df-sdom 8880 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-nn 12135 df-n0 12391 df-z 12478 df-uz 12741

This theorem is referenced by: unben 16825

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