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Theorem seqomlem1 8082
 Description: Lemma for seqω. 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 → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
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.
StepHypRef Expression
1 fveq2 6669 . . 3 (𝑎 = ∅ → (𝑄𝑎) = (𝑄‘∅))
2 id 22 . . . 4 (𝑎 = ∅ → 𝑎 = ∅)
3 2fveq3 6674 . . . 4 (𝑎 = ∅ → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄‘∅)))
42, 3opeq12d 4810 . . 3 (𝑎 = ∅ → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
51, 4eqeq12d 2842 . 2 (𝑎 = ∅ → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩))
6 fveq2 6669 . . 3 (𝑎 = 𝑏 → (𝑄𝑎) = (𝑄𝑏))
7 id 22 . . . 4 (𝑎 = 𝑏𝑎 = 𝑏)
8 2fveq3 6674 . . . 4 (𝑎 = 𝑏 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄𝑏)))
97, 8opeq12d 4810 . . 3 (𝑎 = 𝑏 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
106, 9eqeq12d 2842 . 2 (𝑎 = 𝑏 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩))
11 fveq2 6669 . . 3 (𝑎 = suc 𝑏 → (𝑄𝑎) = (𝑄‘suc 𝑏))
12 id 22 . . . 4 (𝑎 = suc 𝑏𝑎 = suc 𝑏)
13 2fveq3 6674 . . . 4 (𝑎 = suc 𝑏 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄‘suc 𝑏)))
1412, 13opeq12d 4810 . . 3 (𝑎 = suc 𝑏 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩)
1511, 14eqeq12d 2842 . 2 (𝑎 = suc 𝑏 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
16 fveq2 6669 . . 3 (𝑎 = 𝐴 → (𝑄𝑎) = (𝑄𝐴))
17 id 22 . . . 4 (𝑎 = 𝐴𝑎 = 𝐴)
18 2fveq3 6674 . . . 4 (𝑎 = 𝐴 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄𝐴)))
1917, 18opeq12d 4810 . . 3 (𝑎 = 𝐴 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
2016, 19eqeq12d 2842 . 2 (𝑎 = 𝐴 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩))
21 seqomlem.a . . . . 5 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
2221fveq1i 6670 . . . 4 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
23 opex 5353 . . . . 5 ⟨∅, ( I ‘𝐼)⟩ ∈ V
2423rdg0 8053 . . . 4 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
2522, 24eqtri 2849 . . 3 (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩
26 0ex 5208 . . . . . . 7 ∅ ∈ V
27 fvex 6682 . . . . . . 7 ( I ‘𝐼) ∈ V
2826, 27op2nd 7694 . . . . . 6 (2nd ‘⟨∅, ( I ‘𝐼)⟩) = ( I ‘𝐼)
2928eqcomi 2835 . . . . 5 ( I ‘𝐼) = (2nd ‘⟨∅, ( I ‘𝐼)⟩)
3029opeq2i 4806 . . . 4 ⟨∅, ( I ‘𝐼)⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩
31 id 22 . . . 4 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩)
32 fveq2 6669 . . . . 5 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (2nd ‘(𝑄‘∅)) = (2nd ‘⟨∅, ( I ‘𝐼)⟩))
3332opeq2d 4809 . . . 4 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → ⟨∅, (2nd ‘(𝑄‘∅))⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩)
3430, 31, 333eqtr4a 2887 . . 3 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
3525, 34ax-mp 5 . 2 (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩
36 df-ov 7153 . . . . . 6 (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
37 fvex 6682 . . . . . . 7 (2nd ‘(𝑄𝑏)) ∈ V
38 suceq 6255 . . . . . . . . 9 (𝑖 = 𝑏 → suc 𝑖 = suc 𝑏)
39 oveq1 7157 . . . . . . . . 9 (𝑖 = 𝑏 → (𝑖𝐹𝑣) = (𝑏𝐹𝑣))
4038, 39opeq12d 4810 . . . . . . . 8 (𝑖 = 𝑏 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹𝑣)⟩)
41 oveq2 7158 . . . . . . . . 9 (𝑣 = (2nd ‘(𝑄𝑏)) → (𝑏𝐹𝑣) = (𝑏𝐹(2nd ‘(𝑄𝑏))))
4241opeq2d 4809 . . . . . . . 8 (𝑣 = (2nd ‘(𝑄𝑏)) → ⟨suc 𝑏, (𝑏𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
43 eqid 2826 . . . . . . . 8 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
44 opex 5353 . . . . . . . 8 ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ ∈ V
4540, 42, 43, 44ovmpo 7304 . . . . . . 7 ((𝑏 ∈ ω ∧ (2nd ‘(𝑄𝑏)) ∈ V) → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
4637, 45mpan2 687 . . . . . 6 (𝑏 ∈ ω → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
4736, 46syl5eqr 2875 . . . . 5 (𝑏 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
48 fveqeq2 6678 . . . . 5 ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ ↔ ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩))
4947, 48syl5ibrcom 248 . . . 4 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩))
50 vex 3503 . . . . . . . . . 10 𝑏 ∈ V
5150sucex 7519 . . . . . . . . 9 suc 𝑏 ∈ V
52 ovex 7183 . . . . . . . . 9 (𝑏𝐹(2nd ‘(𝑄𝑏))) ∈ V
5351, 52op2nd 7694 . . . . . . . 8 (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩) = (𝑏𝐹(2nd ‘(𝑄𝑏)))
5453eqcomi 2835 . . . . . . 7 (𝑏𝐹(2nd ‘(𝑄𝑏))) = (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
5554a1i 11 . . . . . 6 (𝑏 ∈ ω → (𝑏𝐹(2nd ‘(𝑄𝑏))) = (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩))
5655opeq2d 4809 . . . . 5 (𝑏 ∈ ω → ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ = ⟨suc 𝑏, (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)⟩)
57 id 22 . . . . . 6 (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
58 fveq2 6669 . . . . . . 7 (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ → (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏))) = (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩))
5958opeq2d 4809 . . . . . 6 (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ → ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩ = ⟨suc 𝑏, (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)⟩)
6057, 59eqeq12d 2842 . . . . 5 (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ → (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩ ↔ ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ = ⟨suc 𝑏, (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)⟩))
6156, 60syl5ibrcom 248 . . . 4 (𝑏 ∈ ω → (((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩))
6249, 61syld 47 . . 3 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩))
63 frsuc 8068 . . . . 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 7595 . . . . . . 7 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
6564fvresd 6689 . . . . . 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))
6621fveq1i 6670 . . . . . 6 (𝑄‘suc 𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝑏)
6765, 66syl6eqr 2879 . . . . 5 (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝑏) = (𝑄‘suc 𝑏))
68 fvres 6688 . . . . . . 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))
6921fveq1i 6670 . . . . . . 7 (𝑄𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝑏)
7068, 69syl6eqr 2879 . . . . . 6 (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝑏) = (𝑄𝑏))
7170fveq2d 6673 . . . . 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)⟩)‘(𝑄𝑏)))
7263, 67, 713eqtr3d 2869 . . . 4 (𝑏 ∈ ω → (𝑄‘suc 𝑏) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))
7372fveq2d 6673 . . . . 5 (𝑏 ∈ ω → (2nd ‘(𝑄‘suc 𝑏)) = (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏))))
7473opeq2d 4809 . . . 4 (𝑏 ∈ ω → ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩ = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩)
7572, 74eqeq12d 2842 . . 3 (𝑏 ∈ ω → ((𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩ ↔ ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩))
7662, 75sylibrd 260 . 2 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
775, 10, 15, 20, 35, 76finds 7601 1 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ∈ wcel 2107  Vcvv 3500  ∅c0 4295  ⟨cop 4570   I cid 5458   ↾ cres 5556  suc csuc 6192  ‘cfv 6354  (class class class)co 7150   ∈ cmpo 7152  ωcom 7573  2nd c2nd 7684  reccrdg 8041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042 This theorem is referenced by:  seqomlem2  8083  seqomlem4  8085
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