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Theorem unbenlem 17006

Description: Lemma for unben 17007. (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 12267	. . . . 5 ⊢ ℕ ∈ V
2	1	ssex 5289	. . . 4 ⊢ (𝐴 ⊆ ℕ → 𝐴 ∈ V)
3		1z 12652	. . . . . . . 8 ⊢ 1 ∈ ℤ
4		unbenlem.1	. . . . . . . 8 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)
5	3, 4	om2uzf1oi 14021	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→(ℤ_≥‘1)
6		nnuz 12930	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
7		f1oeq3 6811	. . . . . . . 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 6834	. . . . . 6 ⊢ (𝐺:ω–1-1-onto→ℕ → ^◡𝐺:ℕ–1-1-onto→ω)
11		f1of1 6820	. . . . . 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 6836	. . . . 5 ⊢ ((^◡𝐺:ℕ–1-1→ω ∧ 𝐴 ⊆ ℕ) → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
14	12, 13	mpan 703	. . . 4 ⊢ (𝐴 ⊆ ℕ → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
15		f1oeng 8980	. . . 4 ⊢ ((𝐴 ∈ V ∧ (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴)) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
16	2, 14, 15	syl2anc 596	. . 3 ⊢ (𝐴 ⊆ ℕ → 𝐴 ≈ (^◡𝐺 “ 𝐴))
17	16	adantr 486	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
18		imassrn 6071	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
19		dfdm4 5883	. . . . 5 ⊢ dom 𝐺 = ran ^◡𝐺
20		f1of 6821	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ → 𝐺:ω⟶ℕ)
21	9, 20	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ
22	21	fdmi 6718	. . . . 5 ⊢ dom 𝐺 = ω
23	19, 22	eqtr3i 2787	. . . 4 ⊢ ran ^◡𝐺 = ω
24	18, 23	sseqtri 3982	. . 3 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
25	3, 4	om2uzuzi 14017	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ_≥‘1))
26	25, 6	eleqtrrdi 2873	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℕ)
27		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝐺‘𝑦) → (𝑚 < 𝑛 ↔ (𝐺‘𝑦) < 𝑛))
28	27	rexbidv 3188	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑦) → (∃𝑛 ∈ 𝐴 𝑚 < 𝑛 ↔ ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
29	28	rspcv 3575	. . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛 → ∃𝑛 ∈ 𝐴 (𝐺‘𝑦) < 𝑛))
32		f1ocnv 6834	. . . . . . . . . . . . . . . . 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 6823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:ω–1-1-onto→ℕ → Fun 𝐺)
35	9, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Fun 𝐺
36		funcnvres2 6617	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐺 → ^◡(^◡𝐺 ↾ 𝐴) = (𝐺 ↾ (^◡𝐺 “ 𝐴)))
37		f1oeq1 6809	. . . . . . . . . . . . . . . . 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 6829	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–onto→𝐴)
41		forn 6796	. . . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝑛 ∈ ran (𝐺 ↾ (^◡𝐺 “ 𝐴)) ↔ 𝑛 ∈ 𝐴))
44		f1ofn 6822	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ↾ (^◡𝐺 “ 𝐴)):(^◡𝐺 “ 𝐴)–1-1-onto→𝐴 → (𝐺 ↾ (^◡𝐺 “ 𝐴)) Fn (^◡𝐺 “ 𝐴))
45		fvelrnb 6942	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑛 ∈ 𝐴) → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛)
50		fvres 6901	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = (𝐺‘𝑚))
51	50	eqeq1d 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 ↔ (𝐺‘𝑚) = 𝑛))
52	51	biimpa 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (^◡𝐺 “ 𝐴) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
53	52	adantll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝐺‘𝑚) = 𝑛)
54	24	sseli 3930	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (^◡𝐺 “ 𝐴) → 𝑚 ∈ ω)
55	3, 4	om2uzlt2i 14019	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ ω) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
56	54, 55	sylan2 605	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < (𝐺‘𝑚)))
57		breq2 5111	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑚) = 𝑛 → ((𝐺‘𝑦) < (𝐺‘𝑚) ↔ (𝐺‘𝑦) < 𝑛))
58	56, 57	sylan9bb 519	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ (𝐺‘𝑚) = 𝑛) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < 𝑛))
59	53, 58	syldan 603	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛) → (𝑦 ∈ 𝑚 ↔ (𝐺‘𝑦) < 𝑛))
60	59	biimparc 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺‘𝑦) < 𝑛 ∧ ((𝑦 ∈ ω ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) ∧ ((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛)) → 𝑦 ∈ 𝑚)
61	60	exp44 443	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑦) < 𝑛 → (𝑦 ∈ ω → (𝑚 ∈ (^◡𝐺 “ 𝐴) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → 𝑦 ∈ 𝑚))))
62	61	imp31 423	. . . . . . . . . . . . . 14 ⊢ ((((𝐺‘𝑦) < 𝑛 ∧ 𝑦 ∈ ω) ∧ 𝑚 ∈ (^◡𝐺 “ 𝐴)) → (((𝐺 ↾ (^◡𝐺 “ 𝐴))‘𝑚) = 𝑛 → 𝑦 ∈ 𝑚))
63	62	reximdva 3177	. . . . . . . . . . . . 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 436	. . . . . . . . . . 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 412	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (𝑛 ∈ 𝐴 → ((𝐺‘𝑦) < 𝑛 → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)))
68	67	rexlimdv 3163	. . . . . . . 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 418	. . . . . 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 412	. . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (𝑦 ∈ ω → ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚))
73	72	ralrimiv 3155	. . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚)
74		unbnn3 9642	. . 3 ⊢ (((^◡𝐺 “ 𝐴) ⊆ ω ∧ ∀𝑦 ∈ ω ∃𝑚 ∈ (^◡𝐺 “ 𝐴)𝑦 ∈ 𝑚) → (^◡𝐺 “ 𝐴) ≈ ω)
75	24, 73, 74	sylancr 599	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → (^◡𝐺 “ 𝐴) ≈ ω)
76		entr 9016	. 2 ⊢ ((𝐴 ≈ (^◡𝐺 “ 𝐴) ∧ (^◡𝐺 “ 𝐴) ≈ ω) → 𝐴 ≈ ω)
77	17, 75, 76	syl2anc 596	1 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3078 ∃wrex 3088 Vcvv 3453 ⊆ wss 3902 class class class wbr 5107 ↦ cmpt 5190 ^◡ccnv 5658 dom cdm 5659 ran crn 5660 ↾ cres 5661 “ cima 5662 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 –1-1→wf1 6534 –onto→wfo 6535 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7417 ωcom 7866 reccrdg 8402 ≈ cen 8953 1c1 11129 + caddc 11131 < clt 11271 ℕcn 12261 ℤ_≥cuz 12891

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-n0 12533 df-z 12620 df-uz 12892

This theorem is used by: unben 17007

W3C validator