Theorem uzrdgsuci 13608

Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13604. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)

Hypotheses

Ref	Expression
om2uz.1	⊢ 𝐶 ∈ ℤ
om2uz.2	⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
uzrdg.1	⊢ 𝐴 ∈ V
uzrdg.2	⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
uzrdg.3	⊢ 𝑆 = ran 𝑅

Assertion

Ref	Expression
uzrdgsuci	⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆‘𝐵)))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐶   𝑦,𝐺   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem uzrdgsuci

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		om2uz.1	. . . . . 6 ⊢ 𝐶 ∈ ℤ
2		om2uz.2	. . . . . 6 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
3		uzrdg.1	. . . . . 6 ⊢ 𝐴 ∈ V
4		uzrdg.2	. . . . . 6 ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
5		uzrdg.3	. . . . . 6 ⊢ 𝑆 = ran 𝑅
6	1, 2, 3, 4, 5	uzrdgfni 13606	. . . . 5 ⊢ 𝑆 Fn (ℤ≥‘𝐶)
7		fnfun 6517	. . . . 5 ⊢ (𝑆 Fn (ℤ≥‘𝐶) → Fun 𝑆)
8	6, 7	ax-mp 5	. . . 4 ⊢ Fun 𝑆
9		peano2uz 12570	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
10	1, 2, 3, 4	uzrdglem 13605	. . . . . 6 ⊢ ((𝐵 + 1) ∈ (ℤ≥‘𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ ran 𝑅)
11	9, 10	syl 17	. . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ ran 𝑅)
12	11, 5	eleqtrrdi 2850	. . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆)
13		funopfv 6803	. . . 4 ⊢ (Fun 𝑆 → (⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆 → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))))
14	8, 12, 13	mpsyl 68	. . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1)))))
15	1, 2	om2uzf1oi 13601	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)
16		f1ocnvdm 7137	. . . . . . . 8 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
17	15, 16	mpan 686	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (◡𝐺‘𝐵) ∈ ω)
18		peano2 7711	. . . . . . 7 ⊢ ((◡𝐺‘𝐵) ∈ ω → suc (◡𝐺‘𝐵) ∈ ω)
19	17, 18	syl 17	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → suc (◡𝐺‘𝐵) ∈ ω)
20	1, 2	om2uzsuci 13596	. . . . . . . 8 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝐺‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵)) + 1))
21	17, 20	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐺‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵)) + 1))
22		f1ocnvfv2 7130	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
23	15, 22	mpan 686	. . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
24	23	oveq1d 7270	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ((𝐺‘(◡𝐺‘𝐵)) + 1) = (𝐵 + 1))
25	21, 24	eqtrd 2778	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1))
26		f1ocnvfv 7131	. . . . . . 7 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ suc (◡𝐺‘𝐵) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵)))
27	15, 26	mpan 686	. . . . . 6 ⊢ (suc (◡𝐺‘𝐵) ∈ ω → ((𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵)))
28	19, 25, 27	sylc 65	. . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵))
29	28	fveq2d 6760	. . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑅‘(◡𝐺‘(𝐵 + 1))) = (𝑅‘suc (◡𝐺‘𝐵)))
30	29	fveq2d 6760	. . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
31	14, 30	eqtrd 2778	. 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
32		frsuc 8238	. . . . . . . 8 ⊢ ((◡𝐺‘𝐵) ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵))))
33	4	fveq1i 6757	. . . . . . . 8 ⊢ (𝑅‘suc (◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (◡𝐺‘𝐵))
34	4	fveq1i 6757	. . . . . . . . 9 ⊢ (𝑅‘(◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵))
35	34	fveq2i 6759	. . . . . . . 8 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵)))
36	32, 33, 35	3eqtr4g 2804	. . . . . . 7 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))))
37	1, 2, 3, 4	om2uzrdg 13604	. . . . . . . . 9 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
38	37	fveq2d 6760	. . . . . . . 8 ⊢ ((◡𝐺‘𝐵) ∈ ω → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩))
39		df-ov 7258	. . . . . . . 8 ⊢ ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
40	38, 39	eqtr4di 2797	. . . . . . 7 ⊢ ((◡𝐺‘𝐵) ∈ ω → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
41	36, 40	eqtrd 2778	. . . . . 6 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
42		fvex 6769	. . . . . . 7 ⊢ (𝐺‘(◡𝐺‘𝐵)) ∈ V
43		fvex 6769	. . . . . . 7 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V
44		oveq1 7262	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧 + 1) = ((𝐺‘(◡𝐺‘𝐵)) + 1))
45		oveq1 7262	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤))
46	44, 45	opeq12d 4809	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩)
47		oveq2 7263	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
48	47	opeq2d 4808	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
49		oveq1 7262	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
50		oveq1 7262	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
51	49, 50	opeq12d 4809	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
52		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
53	52	opeq2d 4808	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
54	51, 53	cbvmpov 7348	. . . . . . . 8 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
55		opex 5373	. . . . . . . 8 ⊢ ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ V
56	46, 48, 54, 55	ovmpo 7411	. . . . . . 7 ⊢ (((𝐺‘(◡𝐺‘𝐵)) ∈ V ∧ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V) → ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
57	42, 43, 56	mp2an 688	. . . . . 6 ⊢ ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩
58	41, 57	eqtrdi 2795	. . . . 5 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
59	58	fveq2d 6760	. . . 4 ⊢ ((◡𝐺‘𝐵) ∈ ω → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩))
60		ovex 7288	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵)) + 1) ∈ V
61		ovex 7288	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ V
62	60, 61	op2nd 7813	. . . 4 ⊢ (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))
63	59, 62	eqtrdi 2795	. . 3 ⊢ ((◡𝐺‘𝐵) ∈ ω → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
64	17, 63	syl 17	. 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
65	1, 2, 3, 4	uzrdglem 13605	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
66	65, 5	eleqtrrdi 2850	. . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑆)
67		funopfv 6803	. . . . 5 ⊢ (Fun 𝑆 → (⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑆 → (𝑆‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
68	8, 66, 67	mpsyl 68	. . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵))))
69	68	eqcomd 2744	. . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) = (𝑆‘𝐵))
70	23, 69	oveq12d 7273	. 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = (𝐵𝐹(𝑆‘𝐵)))
71	31, 64, 70	3eqtrd 2782	1 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆‘𝐵)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   ↦ cmpt 5153  ^◡ccnv 5579  ran crn 5581   ↾ cres 5582  suc csuc 6253  Fun wfun 6412   Fn wfn 6413  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  ωcom 7687  2^nd c2nd 7803  reccrdg 8211  1c1 10803   + caddc 10805  ℤcz 12249  ℤ_≥cuz 12511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512

This theorem is referenced by:  seqp1  13664