Theorem uzrdgsuci 12759

Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 12755. (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 12757	. . . . 5 ⊢ 𝑆 Fn (ℤ≥‘𝐶)
7		fnfun 5988	. . . . 5 ⊢ (𝑆 Fn (ℤ≥‘𝐶) → Fun 𝑆)
8	6, 7	ax-mp 5	. . . 4 ⊢ Fun 𝑆
9		peano2uz 11741	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
10	1, 2, 3, 4	uzrdglem 12756	. . . . . 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	syl6eleqr 2712	. . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆)
13		funopfv 6235	. . . 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 12752	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)
16		f1ocnvdm 6540	. . . . . . . 8 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
17	15, 16	mpan 706	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (◡𝐺‘𝐵) ∈ ω)
18		peano2 7086	. . . . . . 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 12747	. . . . . . . 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 6533	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
23	15, 22	mpan 706	. . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
24	23	oveq1d 6665	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ((𝐺‘(◡𝐺‘𝐵)) + 1) = (𝐵 + 1))
25	21, 24	eqtrd 2656	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1))
26		f1ocnvfv 6534	. . . . . . 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 706	. . . . . 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 6195	. . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑅‘(◡𝐺‘(𝐵 + 1))) = (𝑅‘suc (◡𝐺‘𝐵)))
30	29	fveq2d 6195	. . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
31	14, 30	eqtrd 2656	. 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
32		frsuc 7532	. . . . . . . 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 6192	. . . . . . . 8 ⊢ (𝑅‘suc (◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (◡𝐺‘𝐵))
34	4	fveq1i 6192	. . . . . . . . 9 ⊢ (𝑅‘(◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵))
35	34	fveq2i 6194	. . . . . . . 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 2681	. . . . . . 7 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))))
37	1, 2, 3, 4	om2uzrdg 12755	. . . . . . . . 9 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
38	37	fveq2d 6195	. . . . . . . 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 6653	. . . . . . . 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	syl6eqr 2674	. . . . . . 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 2656	. . . . . 6 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
42		fvex 6201	. . . . . . 7 ⊢ (𝐺‘(◡𝐺‘𝐵)) ∈ V
43		fvex 6201	. . . . . . 7 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V
44		oveq1 6657	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧 + 1) = ((𝐺‘(◡𝐺‘𝐵)) + 1))
45		oveq1 6657	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤))
46	44, 45	opeq12d 4410	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩)
47		oveq2 6658	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
48	47	opeq2d 4409	. . . . . . . 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 6657	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
50		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
51	49, 50	opeq12d 4410	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
52		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
53	52	opeq2d 4409	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
54	51, 53	cbvmpt2v 6735	. . . . . . . 8 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
55		opex 4932	. . . . . . . 8 ⊢ ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ V
56	46, 48, 54, 55	ovmpt2 6796	. . . . . . 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 708	. . . . . 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	syl6eq 2672	. . . . 5 ⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
59	58	fveq2d 6195	. . . 4 ⊢ ((◡𝐺‘𝐵) ∈ ω → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩))
60		ovex 6678	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵)) + 1) ∈ V
61		ovex 6678	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ V
62	60, 61	op2nd 7177	. . . 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	syl6eq 2672	. . 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 12756	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
66	65, 5	syl6eleqr 2712	. . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑆)
67		funopfv 6235	. . . . 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 2628	. . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) = (𝑆‘𝐵))
70	23, 69	oveq12d 6668	. 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = (𝐵𝐹(𝑆‘𝐵)))
71	31, 64, 70	3eqtrd 2660	1 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆‘𝐵)))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   ↦ cmpt 4729  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116  suc csuc 5725  Fun wfun 5882   Fn wfn 5883  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  2^nd c2nd 7167  reccrdg 7505  1c1 9937   + caddc 9939  ℤcz 11377  ℤ_≥cuz 11687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688

This theorem is referenced by:  seqp1  12816