Theorem seqomlem1 8381

Description: Lemma for seq_ω. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)

Hypothesis

Ref	Expression
seqomlem.a	⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)

Assertion

Ref	Expression
seqomlem1	⊢ (𝐴 ∈ ω → (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)

Distinct variable groups:   𝑄,𝑖,𝑣   𝐴,𝑖,𝑣   𝑖,𝐹,𝑣

Allowed substitution hints:   𝐼(𝑣,𝑖)

Proof of Theorem seqomlem1

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . 3 ⊢ (𝑎 = ∅ → (𝑄‘𝑎) = (𝑄‘∅))
2		id 22	. . . 4 ⊢ (𝑎 = ∅ → 𝑎 = ∅)
3		2fveq3 6839	. . . 4 ⊢ (𝑎 = ∅ → (2nd ‘(𝑄‘𝑎)) = (2nd ‘(𝑄‘∅)))
4	2, 3	opeq12d 4837	. . 3 ⊢ (𝑎 = ∅ → ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
5	1, 4	eqeq12d 2752	. 2 ⊢ (𝑎 = ∅ → ((𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ ↔ (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩))
6		fveq2 6834	. . 3 ⊢ (𝑎 = 𝑏 → (𝑄‘𝑎) = (𝑄‘𝑏))
7		id 22	. . . 4 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏)
8		2fveq3 6839	. . . 4 ⊢ (𝑎 = 𝑏 → (2nd ‘(𝑄‘𝑎)) = (2nd ‘(𝑄‘𝑏)))
9	7, 8	opeq12d 4837	. . 3 ⊢ (𝑎 = 𝑏 → ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩)
10	6, 9	eqeq12d 2752	. 2 ⊢ (𝑎 = 𝑏 → ((𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ ↔ (𝑄‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩))
11		fveq2 6834	. . 3 ⊢ (𝑎 = suc 𝑏 → (𝑄‘𝑎) = (𝑄‘suc 𝑏))
12		id 22	. . . 4 ⊢ (𝑎 = suc 𝑏 → 𝑎 = suc 𝑏)
13		2fveq3 6839	. . . 4 ⊢ (𝑎 = suc 𝑏 → (2nd ‘(𝑄‘𝑎)) = (2nd ‘(𝑄‘suc 𝑏)))
14	12, 13	opeq12d 4837	. . 3 ⊢ (𝑎 = suc 𝑏 → ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩)
15	11, 14	eqeq12d 2752	. 2 ⊢ (𝑎 = suc 𝑏 → ((𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ ↔ (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
16		fveq2 6834	. . 3 ⊢ (𝑎 = 𝐴 → (𝑄‘𝑎) = (𝑄‘𝐴))
17		id 22	. . . 4 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
18		2fveq3 6839	. . . 4 ⊢ (𝑎 = 𝐴 → (2nd ‘(𝑄‘𝑎)) = (2nd ‘(𝑄‘𝐴)))
19	17, 18	opeq12d 4837	. . 3 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
20	16, 19	eqeq12d 2752	. 2 ⊢ (𝑎 = 𝐴 → ((𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ ↔ (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩))
21		seqomlem.a	. . . . 5 ⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
22	21	fveq1i 6835	. . . 4 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
23		opex 5412	. . . . 5 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ V
24	23	rdg0 8352	. . . 4 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
25	22, 24	eqtri 2759	. . 3 ⊢ (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩
26		0ex 5252	. . . . . . 7 ⊢ ∅ ∈ V
27		fvex 6847	. . . . . . 7 ⊢ ( I ‘𝐼) ∈ V
28	26, 27	op2nd 7942	. . . . . 6 ⊢ (2nd ‘⟨∅, ( I ‘𝐼)⟩) = ( I ‘𝐼)
29	28	eqcomi 2745	. . . . 5 ⊢ ( I ‘𝐼) = (2nd ‘⟨∅, ( I ‘𝐼)⟩)
30	29	opeq2i 4833	. . . 4 ⊢ ⟨∅, ( I ‘𝐼)⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩
31		id 22	. . . 4 ⊢ ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩)
32		fveq2 6834	. . . . 5 ⊢ ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (2nd ‘(𝑄‘∅)) = (2nd ‘⟨∅, ( I ‘𝐼)⟩))
33	32	opeq2d 4836	. . . 4 ⊢ ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → ⟨∅, (2nd ‘(𝑄‘∅))⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩)
34	30, 31, 33	3eqtr4a 2797	. . 3 ⊢ ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
35	25, 34	ax-mp 5	. 2 ⊢ (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩
36		df-ov 7361	. . . . . 6 ⊢ (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝑏))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩)
37		fvex 6847	. . . . . . 7 ⊢ (2nd ‘(𝑄‘𝑏)) ∈ V
38		suceq 6385	. . . . . . . . 9 ⊢ (𝑖 = 𝑏 → suc 𝑖 = suc 𝑏)
39		oveq1 7365	. . . . . . . . 9 ⊢ (𝑖 = 𝑏 → (𝑖𝐹𝑣) = (𝑏𝐹𝑣))
40	38, 39	opeq12d 4837	. . . . . . . 8 ⊢ (𝑖 = 𝑏 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹𝑣)⟩)
41		oveq2 7366	. . . . . . . . 9 ⊢ (𝑣 = (2nd ‘(𝑄‘𝑏)) → (𝑏𝐹𝑣) = (𝑏𝐹(2nd ‘(𝑄‘𝑏))))
42	41	opeq2d 4836	. . . . . . . 8 ⊢ (𝑣 = (2nd ‘(𝑄‘𝑏)) → ⟨suc 𝑏, (𝑏𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)
43		eqid 2736	. . . . . . . 8 ⊢ (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
44		opex 5412	. . . . . . . 8 ⊢ ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ ∈ V
45	40, 42, 43, 44	ovmpo 7518	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (2nd ‘(𝑄‘𝑏)) ∈ V) → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)
46	37, 45	mpan2 691	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)
47	36, 46	eqtr3id 2785	. . . . 5 ⊢ (𝑏 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)
48		fveqeq2 6843	. . . . 5 ⊢ ((𝑄‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩ → (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ ↔ ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩))
49	47, 48	syl5ibrcom 247	. . . 4 ⊢ (𝑏 ∈ ω → ((𝑄‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩))
50		vex 3444	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
51	50	sucex 7751	. . . . . . . . 9 ⊢ suc 𝑏 ∈ V
52		ovex 7391	. . . . . . . . 9 ⊢ (𝑏𝐹(2nd ‘(𝑄‘𝑏))) ∈ V
53	51, 52	op2nd 7942	. . . . . . . 8 ⊢ (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩) = (𝑏𝐹(2nd ‘(𝑄‘𝑏)))
54	53	eqcomi 2745	. . . . . . 7 ⊢ (𝑏𝐹(2nd ‘(𝑄‘𝑏))) = (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)
55	54	a1i 11	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝑏𝐹(2nd ‘(𝑄‘𝑏))) = (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩))
56	55	opeq2d 4836	. . . . 5 ⊢ (𝑏 ∈ ω → ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ = ⟨suc 𝑏, (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)⟩)
57		id 22	. . . . . 6 ⊢ (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)
58		fveq2 6834	. . . . . . 7 ⊢ (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ → (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏))) = (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩))
59	58	opeq2d 4836	. . . . . 6 ⊢ (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ → ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)))⟩ = ⟨suc 𝑏, (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)⟩)
60	57, 59	eqeq12d 2752	. . . . 5 ⊢ (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ → (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)))⟩ ↔ ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ = ⟨suc 𝑏, (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩)⟩))
61	56, 60	syl5ibrcom 247	. . . 4 ⊢ (𝑏 ∈ ω → (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄‘𝑏)))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)))⟩))
62	49, 61	syld 47	. . 3 ⊢ (𝑏 ∈ ω → ((𝑄‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)))⟩))
63		frsuc 8368	. . . . 5 ⊢ (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝑏) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝑏)))
64		peano2 7832	. . . . . . 7 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
65	64	fvresd 6854	. . . . . 6 ⊢ (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝑏))
66	21	fveq1i 6835	. . . . . 6 ⊢ (𝑄‘suc 𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝑏)
67	65, 66	eqtr4di 2789	. . . . 5 ⊢ (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝑏) = (𝑄‘suc 𝑏))
68		fvres 6853	. . . . . . 7 ⊢ (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝑏))
69	21	fveq1i 6835	. . . . . . 7 ⊢ (𝑄‘𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝑏)
70	68, 69	eqtr4di 2789	. . . . . 6 ⊢ (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝑏) = (𝑄‘𝑏))
71	70	fveq2d 6838	. . . . 5 ⊢ (𝑏 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝑏)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)))
72	63, 67, 71	3eqtr3d 2779	. . . 4 ⊢ (𝑏 ∈ ω → (𝑄‘suc 𝑏) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)))
73	72	fveq2d 6838	. . . . 5 ⊢ (𝑏 ∈ ω → (2nd ‘(𝑄‘suc 𝑏)) = (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏))))
74	73	opeq2d 4836	. . . 4 ⊢ (𝑏 ∈ ω → ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩ = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)))⟩)
75	72, 74	eqeq12d 2752	. . 3 ⊢ (𝑏 ∈ ω → ((𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩ ↔ ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)) = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝑏)))⟩))
76	62, 75	sylibrd 259	. 2 ⊢ (𝑏 ∈ ω → ((𝑄‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩ → (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
77	5, 10, 15, 20, 35, 76	finds 7838	1 ⊢ (𝐴 ∈ ω → (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  ⟨cop 4586   I cid 5518   ↾ cres 5626  suc csuc 6319  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  ωcom 7808  2^nd c2nd 7932  reccrdg 8340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  seqomlem2  8382  seqomlem4  8384