relexp0a - Mathbox for Richard Penner

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Theorem relexp0a 44368

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 12506	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 7419	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟1))
3	2	oveq1d 7426	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟1)↑_𝑟0))
4	3	sseq1d 3976	. . . . . 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 7419	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑦))
7	6	oveq1d 7426	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑦)↑_𝑟0))
8	7	sseq1d 3976	. . . . . 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 7419	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟(𝑦 + 1)))
11	10	oveq1d 7426	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0))
12	11	sseq1d 3976	. . . . . 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 7419	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐴↑_𝑟𝑥) = (𝐴↑_𝑟𝑁))
15	14	oveq1d 7426	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴↑_𝑟𝑥)↑_𝑟0) = ((𝐴↑_𝑟𝑁)↑_𝑟0))
16	15	sseq1d 3976	. . . . . 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 15063	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟1) = 𝐴)
19	18	oveq1d 7426	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) = (𝐴↑_𝑟0))
20		ssid 3967	. . . . . 6 ⊢ (𝐴↑_𝑟0) ⊆ (𝐴↑_𝑟0)
21	19, 20	eqsstrdi 3989	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟1)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
22		simp2 1153	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝐴 ∈ 𝑉)
23		simp1 1152	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → 𝑦 ∈ ℕ)
24		relexpsucnnr 15062	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → (𝐴↑_𝑟(𝑦 + 1)) = ((𝐴↑_𝑟𝑦) ∘ 𝐴))
25	24	oveq1d 7426	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℕ) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
26	22, 23, 25	syl2anc 595	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) = (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0))
27		ovex 7444	. . . . . . . . . . . . 13 ⊢ (𝐴↑_𝑟𝑦) ∈ V
28		coexg 7926	. . . . . . . . . . . . 13 ⊢ (((𝐴↑_𝑟𝑦) ∈ V ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
29	27, 28	mpan 702	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∈ V)
30		relexp0g 15059	. . . . . . . . . . . 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 5966	. . . . . . . . . . . . 13 ⊢ dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33		rncoss 5968	. . . . . . . . . . . . 13 ⊢ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴↑_𝑟𝑦)
34		unss12 4149	. . . . . . . . . . . . 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 704	. . . . . . . . . . . 12 ⊢ (dom ((𝐴↑_𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴↑_𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))
36		ssres2 6004	. . . . . . . . . . . 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 3989	. . . . . . . . . 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 5993	. . . . . . . . . . 11 ⊢ ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦)))
41		ssun1 4139	. . . . . . . . . . . . . . 15 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42		ssres2 6004	. . . . . . . . . . . . . . 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 15059	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (𝐴↑_𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
45	43, 44	sseqtrrid 3988	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
46	45	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴↑_𝑟0))
47		ssun2 4140	. . . . . . . . . . . . . . 15 ⊢ ran (𝐴↑_𝑟𝑦) ⊆ (dom (𝐴↑_𝑟𝑦) ∪ ran (𝐴↑_𝑟𝑦))
48		ssres2 6004	. . . . . . . . . . . . . . 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 15059	. . . . . . . . . . . . . . 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 3994	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ ((𝐴↑_𝑟𝑦)↑_𝑟0)
53		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
54	52, 53	sstrid 3956	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ ran (𝐴↑_𝑟𝑦)) ⊆ (𝐴↑_𝑟0))
55	46, 54	unssd 4153	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
56	40, 55	eqsstrid 3983	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
57	56	3adant1 1146	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴↑_𝑟𝑦))) ⊆ (𝐴↑_𝑟0))
58	39, 57	sstrd 3955	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → (((𝐴↑_𝑟𝑦) ∘ 𝐴)↑_𝑟0) ⊆ (𝐴↑_𝑟0))
59	26, 58	eqsstrd 3979	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 ∈ 𝑉 ∧ ((𝐴↑_𝑟𝑦)↑_𝑟0) ⊆ (𝐴↑_𝑟0)) → ((𝐴↑_𝑟(𝑦 + 1))↑_𝑟0) ⊆ (𝐴↑_𝑟0))
60	59	3exp 1135	. . . . . 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 12251	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
63		oveq2 7419	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝐴↑_𝑟𝑁) = (𝐴↑_𝑟0))
64	63	oveq1d 7426	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝐴↑_𝑟𝑁)↑_𝑟0) = ((𝐴↑_𝑟0)↑_𝑟0))
65		relexp0idm 44367	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟0)↑_𝑟0) = (𝐴↑_𝑟0))
66	64, 65	sylan9eq 2824	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝐴 ∈ 𝑉) → ((𝐴↑_𝑟𝑁)↑_𝑟0) = (𝐴↑_𝑟0))
67		eqimss 4003	. . . . . 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 417	. . . 4 ⊢ (𝑁 = 0 → (𝐴 ∈ 𝑉 → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0)))
70	62, 69	jaoi 870	. . 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 412	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝐴↑_𝑟𝑁)↑_𝑟0) ⊆ (𝐴↑_𝑟0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ∪ cun 3911 ⊆ wss 3913 I cid 5556 dom cdm 5662 ran crn 5663 ↾ cres 5664 ∘ ccom 5666 (class class class)co 7411 0cc0 11100 1c1 11101 + caddc 11103 ℕcn 12233 ℕ₀cn0 12504 ↑_𝑟crelexp 15056

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-n0 12505 df-z 12592 df-uz 12863 df-seq 14038 df-relexp 15057

This theorem is referenced by: (None)

W3C validator