Theorem uzrdgsuci 14001

Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13997. (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 13999	. . . . 5 ⊢ 𝑆 Fn (ℤ≥‘𝐶)
7		fnfun 6668	. . . . 5 ⊢ (𝑆 Fn (ℤ≥‘𝐶) → Fun 𝑆)
8	6, 7	ax-mp 5	. . . 4 ⊢ Fun 𝑆
9		peano2uz 12943	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
10	1, 2, 3, 4	uzrdglem 13998	. . . . . 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 2852	. . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆)
13		funopfv 6958	. . . 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 13994	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)
16		f1ocnvdm 7305	. . . . . . . 8 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
17	15, 16	mpan 690	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (◡𝐺‘𝐵) ∈ ω)
18		peano2 7912	. . . . . . 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 13989	. . . . . . . 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 7297	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
23	15, 22	mpan 690	. . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
24	23	oveq1d 7446	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ((𝐺‘(◡𝐺‘𝐵)) + 1) = (𝐵 + 1))
25	21, 24	eqtrd 2777	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1))
26		f1ocnvfv 7298	. . . . . . 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 690	. . . . . 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 6910	. . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑅‘(◡𝐺‘(𝐵 + 1))) = (𝑅‘suc (◡𝐺‘𝐵)))
30	29	fveq2d 6910	. . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
31	14, 30	eqtrd 2777	. 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
32		frsuc 8477	. . . . . . . 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 6907	. . . . . . . 8 ⊢ (𝑅‘suc (◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (◡𝐺‘𝐵))
34	4	fveq1i 6907	. . . . . . . . 9 ⊢ (𝑅‘(◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵))
35	34	fveq2i 6909	. . . . . . . 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 2802	. . . . . . 7 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))))
37	1, 2, 3, 4	om2uzrdg 13997	. . . . . . . . 9 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
38	37	fveq2d 6910	. . . . . . . 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 7434	. . . . . . . 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 2795	. . . . . . 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 2777	. . . . . 6 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
42		fvex 6919	. . . . . . 7 ⊢ (𝐺‘(◡𝐺‘𝐵)) ∈ V
43		fvex 6919	. . . . . . 7 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V
44		oveq1 7438	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧 + 1) = ((𝐺‘(◡𝐺‘𝐵)) + 1))
45		oveq1 7438	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤))
46	44, 45	opeq12d 4881	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩)
47		oveq2 7439	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
48	47	opeq2d 4880	. . . . . . . 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 7438	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
50		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
51	49, 50	opeq12d 4881	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
52		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
53	52	opeq2d 4880	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
54	51, 53	cbvmpov 7528	. . . . . . . 8 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
55		opex 5469	. . . . . . . 8 ⊢ ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ V
56	46, 48, 54, 55	ovmpo 7593	. . . . . . 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 692	. . . . . 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 2793	. . . . 5 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
59	58	fveq2d 6910	. . . 4 ⊢ ((◡𝐺‘𝐵) ∈ ω → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩))
60		ovex 7464	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵)) + 1) ∈ V
61		ovex 7464	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ V
62	60, 61	op2nd 8023	. . . 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 2793	. . 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 13998	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
66	65, 5	eleqtrrdi 2852	. . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑆)
67		funopfv 6958	. . . . 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 2743	. . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) = (𝑆‘𝐵))
70	23, 69	oveq12d 7449	. 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = (𝐵𝐹(𝑆‘𝐵)))
71	31, 64, 70	3eqtrd 2781	1 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆‘𝐵)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632   ↦ cmpt 5225  ^◡ccnv 5684  ran crn 5686   ↾ cres 5687  suc csuc 6386  Fun wfun 6555   Fn wfn 6556  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  ωcom 7887  2^nd c2nd 8013  reccrdg 8449  1c1 11156   + caddc 11158  ℤcz 12613  ℤ_≥cuz 12878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879

This theorem is referenced by:  seqp1  14057