relexp0a - Mathbox for Richard Penner

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Theorem relexp0a 44621

Description: Absorption law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)

**Assertion**
Ref	Expression
relexp0a	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))

Proof of Theorem relexp0a Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 12555	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 7424	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟1))
3	2	oveq1d 7431	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟1)↑_𝑟0))
4	3	sseq1d 3962	. . . . . 6 ⊢ (𝑥 = 1 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
5	4	imbi2d 343	. . . . 5 ⊢ (𝑥 = 1 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
6		oveq2 7424	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑦))
7	6	oveq1d 7431	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑦)↑_𝑟0))
8	7	sseq1d 3962	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
9	8	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
10		oveq2 7424	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟(𝑦 + 1)))
11	10	oveq1d 7431	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0))
12	11	sseq1d 3962	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
13	12	imbi2d 343	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
14		oveq2 7424	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑁))
15	14	oveq1d 7431	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑁)↑_𝑟0))
16	15	sseq1d 3962	. . . . . 6 ⊢ (𝑥 = 𝑁 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
17	16	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
18		relexp1g 15124	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟1) = 𝐴)
19	18	oveq1d 7431	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) = (𝐴↑_𝑟0))
20		ssid 3953	. . . . . 6 ⊢ (𝐴↑_𝑟0) ⊆ (𝐴↑_𝑟0)
21	19, 20	eqsstrdi 3975	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
22		simp2 1155	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝐴 ∈ 𝑉)
23		simp1 1154	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝑦 ∈ ℕ)
24		relexpsucnnr 15123	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝐴↑_𝑟(𝑦 + 1)) = ((𝐴↑_𝑟𝑦) ∘ 𝐴))
25	24	oveq1d 7431	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
26	22, 23, 25	syl2anc 596	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
27		ovex 7449	. . . . . . . . . . . . 13 ⊢ (𝐴↑_𝑟𝑦) ∈ V
28		coexg 7932	. . . . . . . . . . . . 13 ⊢ (((𝐴↑_𝑟𝑦) ∈ V ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
29	27, 28	mpan 703	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
30		relexp0g 15120	. . . . . . . . . . . 12 ⊢ (((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) = ( I ↾ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴))))
31	29, 30	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) = ( I ↾ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴))))
32		dmcoss 5961	. . . . . . . . . . . . 13 ⊢ dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33		rncoss 5963	. . . . . . . . . . . . 13 ⊢ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴↑_𝑟𝑦)
34		unss12 4134	. . . . . . . . . . . . 13 ⊢ ((dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴 ∧ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴↑_𝑟𝑦)) → (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦)))
35	32, 33, 34	mp2an 705	. . . . . . . . . . . 12 ⊢ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))
36		ssres2 5999	. . . . . . . . . . . 12 ⊢ ((dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦)) → ( I ↾ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴))) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))))
37	35, 36	ax-mp 5	. . . . . . . . . . 11 ⊢ ( I ↾ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴))) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦)))
38	31, 37	eqsstrdi 3975	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))))
39	22, 38	syl 18	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))))
40		resundi 5988	. . . . . . . . . . 11 ⊢ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦)))
41		ssun1 4124	. . . . . . . . . . . . . . 15 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42		ssres2 5999	. . . . . . . . . . . . . . 15 ⊢ (dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) → ( I ↾ dom 𝐴) ⊆ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
43	41, 42	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( I ↾ dom 𝐴) ⊆ ( I ↾ (dom 𝐴 ∪ ran 𝐴))
44		relexp0g 15120	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
45	43, 44	sseqtrrid 3974	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
46	45	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
47		ssun2 4125	. . . . . . . . . . . . . . 15 ⊢ ran (𝐴↑_𝑟𝑦) ⊆ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦))
48		ssres2 5999	. . . . . . . . . . . . . . 15 ⊢ (ran (𝐴↑_𝑟𝑦) ⊆ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦)) → ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ ( I ↾ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦))))
49	47, 48	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ ( I ↾ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦)))
50		relexp0g 15120	. . . . . . . . . . . . . . 15 ⊢ ((𝐴↑_𝑟𝑦) ∈ V → ((𝐴↑_𝑟𝑦)↑_𝑟0) = ( I ↾ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦))))
51	27, 50	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝐴↑_𝑟𝑦)↑_𝑟0) = ( I ↾ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦)))
52	49, 51	sseqtrri 3980	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ ((𝐴↑_𝑟𝑦)↑_𝑟0)
53		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
54	52, 53	sstrid 3942	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ (𝐴↑_𝑟0))
55	46, 54	unssd 4138	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
56	40, 55	eqsstrid 3969	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
57	56	3adant1 1148	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
58	39, 57	sstrd 3941	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
59	26, 58	eqsstrd 3965	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))
60	59	3exp 1137	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (𝐴 ∈ 𝑉 → (((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
61	60	a2d 30	. . . . 5 ⊢ (𝑦 ∈ ℕ → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
62	5, 9, 13, 17, 21, 61	nnind 12300	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
63		oveq2 7424	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝐴↑_𝑟𝑁) = (𝐴↑_𝑟0))
64	63	oveq1d 7431	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝐴↑_𝑟𝑁)↑_𝑟0) = ((𝐴↑_𝑟0)↑_𝑟0))
65		relexp0idm 44620	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟0)↑_𝑟0) = (𝐴↑_𝑟0))
66	64, 65	sylan9eq 2815	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑁)↑_𝑟0) = (𝐴↑_𝑟0))
67		eqimss 3989	. . . . . 6 ⊢ (((𝐴↑_𝑟𝑁)↑_𝑟0) = (𝐴↑_𝑟0) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
68	66, 67	syl 18	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
69	68	ex 418	. . . 4 ⊢ (𝑁 = 0 → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
70	62, 69	jaoi 871	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
71	1, 70	sylbi 220	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
72	71	impcom 413	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ∪ cun 3897 ⊆ wss 3899 I cid 5549 dom cdm 5655 ran crn 5656 ↾ cres 5657 ∘ ccom 5659 (class class class)co 7416 0cc0 11149 1c1 11150 + caddc 11152 ℕcn 12282 ℕ₀cn0 12553 ↑_𝑟crelexp 15117

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-er 8703 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-nn 12283 df-n0 12554 df-z 12641 df-uz 12913 df-seq 14091 df-relexp 15118

This theorem is used by: (None)

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