relexp0a - Mathbox for Richard Penner

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Theorem relexp0a 43678

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 7456	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟1))
3	2	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟1)↑_𝑟0))
4	3	sseq1d 4040	. . . . . 6 ⊢ (𝑥 = 1 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
5	4	imbi2d 340	. . . . 5 ⊢ (𝑥 = 1 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
6		oveq2 7456	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑦))
7	6	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑦)↑_𝑟0))
8	7	sseq1d 4040	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
9	8	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
10		oveq2 7456	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟(𝑦 + 1)))
11	10	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0))
12	11	sseq1d 4040	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
13	12	imbi2d 340	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
14		oveq2 7456	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑁))
15	14	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑁)↑_𝑟0))
16	15	sseq1d 4040	. . . . . 6 ⊢ (𝑥 = 𝑁 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
17	16	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
18		relexp1g 15075	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟1) = 𝐴)
19	18	oveq1d 7463	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) = (𝐴↑_𝑟0))
20		ssid 4031	. . . . . 6 ⊢ (𝐴↑_𝑟0) ⊆ (𝐴↑_𝑟0)
21	19, 20	eqsstrdi 4063	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
22		simp2 1137	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝐴 ∈ 𝑉)
23		simp1 1136	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝑦 ∈ ℕ)
24		relexpsucnnr 15074	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝐴↑_𝑟(𝑦 + 1)) = ((𝐴↑_𝑟𝑦) ∘ 𝐴))
25	24	oveq1d 7463	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
26	22, 23, 25	syl2anc 583	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
27		ovex 7481	. . . . . . . . . . . . 13 ⊢ (𝐴↑_𝑟𝑦) ∈ V
28		coexg 7969	. . . . . . . . . . . . 13 ⊢ (((𝐴↑_𝑟𝑦) ∈ V ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
29	27, 28	mpan 689	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
30		relexp0g 15071	. . . . . . . . . . . 12 ⊢ (((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) = ( I ↾ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴))))
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) = ( I ↾ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴))))
32		dmcoss 5997	. . . . . . . . . . . . 13 ⊢ dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33		rncoss 5998	. . . . . . . . . . . . 13 ⊢ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴↑_𝑟𝑦)
34		unss12 4211	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))
36		ssres2 6034	. . . . . . . . . . . 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 4063	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))))
39	22, 38	syl 17	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))))
40		resundi 6023	. . . . . . . . . . 11 ⊢ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦)))
41		ssun1 4201	. . . . . . . . . . . . . . 15 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42		ssres2 6034	. . . . . . . . . . . . . . 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 15071	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
45	43, 44	sseqtrrid 4062	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
47		ssun2 4202	. . . . . . . . . . . . . . 15 ⊢ ran (𝐴↑_𝑟𝑦) ⊆ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦))
48		ssres2 6034	. . . . . . . . . . . . . . 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 15071	. . . . . . . . . . . . . . 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 4046	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ ((𝐴↑_𝑟𝑦)↑_𝑟0)
53		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
54	52, 53	sstrid 4020	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ (𝐴↑_𝑟0))
55	46, 54	unssd 4215	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
56	40, 55	eqsstrid 4057	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
57	56	3adant1 1130	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
58	39, 57	sstrd 4019	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
59	26, 58	eqsstrd 4047	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))
60	59	3exp 1119	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (𝐴 ∈ 𝑉 → (((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
61	60	a2d 29	. . . . 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 12311	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
63		oveq2 7456	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝐴↑_𝑟𝑁) = (𝐴↑_𝑟0))
64	63	oveq1d 7463	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝐴↑_𝑟𝑁)↑_𝑟0) = ((𝐴↑_𝑟0)↑_𝑟0))
65		relexp0idm 43677	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟0)↑_𝑟0) = (𝐴↑_𝑟0))
66	64, 65	sylan9eq 2800	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑁)↑_𝑟0) = (𝐴↑_𝑟0))
67		eqimss 4067	. . . . . 6 ⊢ (((𝐴↑_𝑟𝑁)↑_𝑟0) = (𝐴↑_𝑟0) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
68	66, 67	syl 17	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
69	68	ex 412	. . . 4 ⊢ (𝑁 = 0 → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
70	62, 69	jaoi 856	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
71	1, 70	sylbi 217	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
72	71	impcom 407	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ∪ cun 3974 ⊆ wss 3976 I cid 5592 dom cdm 5700 ran crn 5701 ↾ cres 5702 ∘ ccom 5704 (class class class)co 7448 0cc0 11184 1c1 11185 + caddc 11187 ℕcn 12293 ℕ₀cn0 12553 ↑_𝑟crelexp 15068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-seq 14053 df-relexp 15069

This theorem is referenced by: (None)

W3C validator