relexp0a - Mathbox for Richard Penner

	Mathbox for Richard Penner		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > relexp0a		Structured version Visualization version GIF version

Theorem relexp0a 41324

Description: Absorbtion 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 12235	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟1))
3	2	oveq1d 7290	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟1)↑_𝑟0))
4	3	sseq1d 3952	. . . . . 6 ⊢ (𝑥 = 1 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
5	4	imbi2d 341	. . . . 5 ⊢ (𝑥 = 1 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
6		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑦))
7	6	oveq1d 7290	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑦)↑_𝑟0))
8	7	sseq1d 3952	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
9	8	imbi2d 341	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
10		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟(𝑦 + 1)))
11	10	oveq1d 7290	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0))
12	11	sseq1d 3952	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
13	12	imbi2d 341	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
14		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑁))
15	14	oveq1d 7290	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑁)↑_𝑟0))
16	15	sseq1d 3952	. . . . . 6 ⊢ (𝑥 = 𝑁 → (((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0) ↔ ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
17	16	imbi2d 341	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑥)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) ↔ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))))
18		relexp1g 14737	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟1) = 𝐴)
19	18	oveq1d 7290	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) = (𝐴↑_𝑟0))
20		ssid 3943	. . . . . 6 ⊢ (𝐴↑_𝑟0) ⊆ (𝐴↑_𝑟0)
21	19, 20	eqsstrdi 3975	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
22		simp2 1136	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝐴 ∈ 𝑉)
23		simp1 1135	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝑦 ∈ ℕ)
24		relexpsucnnr 14736	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝐴↑_𝑟(𝑦 + 1)) = ((𝐴↑_𝑟𝑦) ∘ 𝐴))
25	24	oveq1d 7290	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
26	22, 23, 25	syl2anc 584	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
27		ovex 7308	. . . . . . . . . . . . 13 ⊢ (𝐴↑_𝑟𝑦) ∈ V
28		coexg 7776	. . . . . . . . . . . . 13 ⊢ (((𝐴↑_𝑟𝑦) ∈ V ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
29	27, 28	mpan 687	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
30		relexp0g 14733	. . . . . . . . . . . 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 5880	. . . . . . . . . . . . 13 ⊢ dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33		rncoss 5881	. . . . . . . . . . . . 13 ⊢ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴↑_𝑟𝑦)
34		unss12 4116	. . . . . . . . . . . . 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 689	. . . . . . . . . . . 12 ⊢ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))
36		ssres2 5919	. . . . . . . . . . . 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 17	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))))
40		resundi 5905	. . . . . . . . . . 11 ⊢ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦)))
41		ssun1 4106	. . . . . . . . . . . . . . 15 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42		ssres2 5919	. . . . . . . . . . . . . . 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 14733	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
47		ssun2 4107	. . . . . . . . . . . . . . 15 ⊢ ran (𝐴↑_𝑟𝑦) ⊆ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦))
48		ssres2 5919	. . . . . . . . . . . . . . 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 14733	. . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ ((𝐴↑_𝑟𝑦)↑_𝑟0)
53		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
54	52, 53	sstrid 3932	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ (𝐴↑_𝑟0))
55	46, 54	unssd 4120	. . . . . . . . . . 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 1129	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
58	39, 57	sstrd 3931	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
59	26, 58	eqsstrd 3959	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))
60	59	3exp 1118	. . . . . 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 11991	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
63		oveq2 7283	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝐴↑_𝑟𝑁) = (𝐴↑_𝑟0))
64	63	oveq1d 7290	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝐴↑_𝑟𝑁)↑_𝑟0) = ((𝐴↑_𝑟0)↑_𝑟0))
65		relexp0idm 41323	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟0)↑_𝑟0) = (𝐴↑_𝑟0))
66	64, 65	sylan9eq 2798	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑁)↑_𝑟0) = (𝐴↑_𝑟0))
67		eqimss 3977	. . . . . 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 413	. . . 4 ⊢ (𝑁 = 0 → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
70	62, 69	jaoi 854	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
71	1, 70	sylbi 216	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
72	71	impcom 408	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∪ cun 3885 ⊆ wss 3887 I cid 5488 dom cdm 5589 ran crn 5590 ↾ cres 5591 ∘ ccom 5593 (class class class)co 7275 0cc0 10871 1c1 10872 + caddc 10874 ℕcn 11973 ℕ₀cn0 12233 ↑_𝑟crelexp 14730

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-seq 13722 df-relexp 14731

This theorem is referenced by: (None)

W3C validator