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 40054

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 11893	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 7158	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟1))
3	2	oveq1d 7165	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟1)↑_𝑟0))
4	3	sseq1d 3998	. . . . . 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 7158	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑦))
7	6	oveq1d 7165	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑦)↑_𝑟0))
8	7	sseq1d 3998	. . . . . 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 7158	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟(𝑦 + 1)))
11	10	oveq1d 7165	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0))
12	11	sseq1d 3998	. . . . . 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 7158	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑁))
15	14	oveq1d 7165	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑁)↑_𝑟0))
16	15	sseq1d 3998	. . . . . 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 14379	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟1) = 𝐴)
19	18	oveq1d 7165	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) = (𝐴↑_𝑟0))
20		ssid 3989	. . . . . 6 ⊢ (𝐴↑_𝑟0) ⊆ (𝐴↑_𝑟0)
21	19, 20	eqsstrdi 4021	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
22		simp2 1133	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝐴 ∈ 𝑉)
23		simp1 1132	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝑦 ∈ ℕ)
24		relexpsucnnr 14378	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝐴↑_𝑟(𝑦 + 1)) = ((𝐴↑_𝑟𝑦) ∘ 𝐴))
25	24	oveq1d 7165	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
26	22, 23, 25	syl2anc 586	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
27		ovex 7183	. . . . . . . . . . . . 13 ⊢ (𝐴↑_𝑟𝑦) ∈ V
28		coexg 7628	. . . . . . . . . . . . 13 ⊢ (((𝐴↑_𝑟𝑦) ∈ V ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
29	27, 28	mpan 688	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
30		relexp0g 14375	. . . . . . . . . . . 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 5837	. . . . . . . . . . . . 13 ⊢ dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33		rncoss 5838	. . . . . . . . . . . . 13 ⊢ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴↑_𝑟𝑦)
34		unss12 4158	. . . . . . . . . . . . 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 690	. . . . . . . . . . . 12 ⊢ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))
36		ssres2 5876	. . . . . . . . . . . 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 4021	. . . . . . . . . 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 5862	. . . . . . . . . . 11 ⊢ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦)))
41		ssun1 4148	. . . . . . . . . . . . . . 15 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42		ssres2 5876	. . . . . . . . . . . . . . 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 14375	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
45	43, 44	sseqtrrid 4020	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
46	45	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
47		ssun2 4149	. . . . . . . . . . . . . . 15 ⊢ ran (𝐴↑_𝑟𝑦) ⊆ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦))
48		ssres2 5876	. . . . . . . . . . . . . . 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 14375	. . . . . . . . . . . . . . 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 4004	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ ((𝐴↑_𝑟𝑦)↑_𝑟0)
53		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
54	52, 53	sstrid 3978	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ (𝐴↑_𝑟0))
55	46, 54	unssd 4162	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
56	40, 55	eqsstrid 4015	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
57	56	3adant1 1126	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
58	39, 57	sstrd 3977	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
59	26, 58	eqsstrd 4005	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))
60	59	3exp 1115	. . . . . 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 11650	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
63		oveq2 7158	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝐴↑_𝑟𝑁) = (𝐴↑_𝑟0))
64	63	oveq1d 7165	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝐴↑_𝑟𝑁)↑_𝑟0) = ((𝐴↑_𝑟0)↑_𝑟0))
65		relexp0idm 40053	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟0)↑_𝑟0) = (𝐴↑_𝑟0))
66	64, 65	sylan9eq 2876	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑁)↑_𝑟0) = (𝐴↑_𝑟0))
67		eqimss 4023	. . . . . 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 415	. . . 4 ⊢ (𝑁 = 0 → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
70	62, 69	jaoi 853	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
71	1, 70	sylbi 219	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
72	71	impcom 410	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∪ cun 3934 ⊆ wss 3936 I cid 5454 dom cdm 5550 ran crn 5551 ↾ cres 5552 ∘ ccom 5554 (class class class)co 7150 0cc0 10531 1c1 10532 + caddc 10534 ℕcn 11632 ℕ₀cn0 11891 ↑_𝑟crelexp 14373

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-relexp 14374

This theorem is referenced by: (None)

W3C validator