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Theorem unbenlem 16880

Description: Lemma for unben 16881. (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 12251	. . . . 5 ⊢ ℕ ∈ V
2	1	ssex 5322	. . . 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 13954	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→(ℤ_≥‘1)
6		nnuz 12898	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
7		f1oeq3 6828	. . . . . . . 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 230	. . . . . 6 ⊢ 𝐺:ω–1-1-onto→ℕ
10		f1ocnv 6850	. . . . . 6 ⊢ (𝐺:ω–1-1-onto→ℕ → ^◡𝐺:ℕ–1-1-onto→ω)
11		f1of1 6837	. . . . . 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 6852	. . . . 5 ⊢ ((^◡𝐺:ℕ–1-1→ω ∧ 𝐴 ⊆ ℕ) → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
14	12, 13	mpan 688	. . . 4 ⊢ (𝐴 ⊆ ℕ → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
15		f1oeng 8992	. . . 4 ⊢ ((𝐴 ∈ V ∧ (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴)) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
16	2, 14, 15	syl2anc 582	. . 3 ⊢ (𝐴 ⊆ ℕ → 𝐴 ≈ (^◡𝐺 “ 𝐴))
17	16	adantr 479	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
18		imassrn 6075	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
19		dfdm4 5898	. . . . 5 ⊢ dom 𝐺 = ran ^◡𝐺
20		f1of 6838	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ → 𝐺:ω⟶ℕ)
21	9, 20	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ
22	21	fdmi 6734	. . . . 5 ⊢ dom 𝐺 = ω
23	19, 22	eqtr3i 2755	. . . 4 ⊢ ran ^◡𝐺 = ω
24	18, 23	sseqtri 4013	. . 3 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
25	3, 4	om2uzuzi 13950	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ_≥‘1))
26	25, 6	eleqtrrdi 2836	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℕ)
27		breq1 5152	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝐺‘𝑦) → (𝑚 < 𝑛 ↔ (𝐺‘𝑦) < 𝑛))
28	27	rexbidv 3168	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑦) → (∃𝑛 ∈ 𝐴 𝑚 < 𝑛 ↔ ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
29	28	rspcv 3602	. . . . . . . . . 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 479	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
32		f1ocnv 6850	. . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ω–1-1-onto→ℕ → Fun 𝐺)
35	9, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Fun 𝐺
36		funcnvres2 6634	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐺 → ^◡(^◡𝐺 ↾ 𝐴) = (𝐺 ↾ (^◡𝐺 “ 𝐴)))
37		f1oeq1 6826	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℕ → (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴)
40		f1ofo 6845	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–onto→𝐴)
41		forn 6813	. . . . . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝑛 ∈ ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) ↔ 𝑛 ∈ 𝐴))
44		f1ofn 6839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)) Fn (^◡𝐺 “ 𝐴))
45		fvelrnb 6958	. . . . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑛 ∈ 𝐴) → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛)
50		fvres 6915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = (𝐺‘𝑚))
51	50	eqeq1d 2727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 ↔ (𝐺‘𝑚) = 𝑛))
52	51	biimpa 475	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (^◡𝐺 “ 𝐴) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
53	52	adantll 712	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
54	24	sseli 3972	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → 𝑚 ∈ ω)
55	3, 4	om2uzlt2i 13952	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ ω) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
56	54, 55	sylan2 591	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
57		breq2 5153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑚) = 𝑛 → ((𝐺‘𝑦) < (𝐺‘𝑚) ↔ (𝐺‘𝑦) < 𝑛))
58	56, 57	sylan9bb 508	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ (𝐺‘𝑚) = 𝑛) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < 𝑛))
59	53, 58	syldan 589	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < 𝑛))
60	59	biimparc 478	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺‘𝑦) < 𝑛 ∧ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛)) → 𝑦 ∈ 𝑚)
61	60	exp44 436	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑦) < 𝑛 → (𝑦 ∈ ω → (𝑚 ∈ (^◡𝐺 “ 𝐴) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → 𝑦 ∈ 𝑚))))
62	61	imp31 416	. . . . . . . . . . . . . 14 ⊢ ((((𝐺‘𝑦) < 𝑛 ∧ 𝑦 ∈ ω) ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → 𝑦 ∈ 𝑚))
63	62	reximdva 3157	. . . . . . . . . . . . 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 429	. . . . . . . . . . 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 405	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (𝑛 ∈ 𝐴 → ((𝐺‘𝑦) < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)))
68	67	rexlimdv 3142	. . . . . . . 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 411	. . . . . 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 405	. . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (𝑦 ∈ ω → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
73	72	ralrimiv 3134	. . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)
74		unbnn3 9684	. . 3 ⊢ (((^◡𝐺 “ 𝐴) ⊆ ω ∧ ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚) → (^◡𝐺 “ 𝐴) ≈ ω)
75	24, 73, 74	sylancr 585	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (^◡𝐺 “ 𝐴) ≈ ω)
76		entr 9027	. 2 ⊢ ((𝐴 ≈ (^◡𝐺 “ 𝐴) ∧ (^◡𝐺 “ 𝐴) ≈ ω) → 𝐴 ≈ ω)
77	17, 75, 76	syl2anc 582	1 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 Vcvv 3461 ⊆ wss 3944 class class class wbr 5149 ↦ cmpt 5232 ^◡ccnv 5677 dom cdm 5678 ran crn 5679 ↾ cres 5680 “ cima 5681 Fun wfun 6543 Fn wfn 6544 ⟶wf 6545 –1-1→wf1 6546 –onto→wfo 6547 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ωcom 7871 reccrdg 8430 ≈ cen 8961 1c1 11141 + caddc 11143 < clt 11280 ℕcn 12245 ℤ_≥cuz 12855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856

This theorem is referenced by: unben 16881

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