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Theorem seqomlem1 8445
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 6888 . . 3 (𝑎 = ∅ → (𝑄𝑎) = (𝑄‘∅))
2 id 22 . . . 4 (𝑎 = ∅ → 𝑎 = ∅)
3 2fveq3 6893 . . . 4 (𝑎 = ∅ → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄‘∅)))
42, 3opeq12d 4880 . . 3 (𝑎 = ∅ → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
51, 4eqeq12d 2749 . 2 (𝑎 = ∅ → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩))
6 fveq2 6888 . . 3 (𝑎 = 𝑏 → (𝑄𝑎) = (𝑄𝑏))
7 id 22 . . . 4 (𝑎 = 𝑏𝑎 = 𝑏)
8 2fveq3 6893 . . . 4 (𝑎 = 𝑏 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄𝑏)))
97, 8opeq12d 4880 . . 3 (𝑎 = 𝑏 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
106, 9eqeq12d 2749 . 2 (𝑎 = 𝑏 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩))
11 fveq2 6888 . . 3 (𝑎 = suc 𝑏 → (𝑄𝑎) = (𝑄‘suc 𝑏))
12 id 22 . . . 4 (𝑎 = suc 𝑏𝑎 = suc 𝑏)
13 2fveq3 6893 . . . 4 (𝑎 = suc 𝑏 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄‘suc 𝑏)))
1412, 13opeq12d 4880 . . 3 (𝑎 = suc 𝑏 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩)
1511, 14eqeq12d 2749 . 2 (𝑎 = suc 𝑏 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
16 fveq2 6888 . . 3 (𝑎 = 𝐴 → (𝑄𝑎) = (𝑄𝐴))
17 id 22 . . . 4 (𝑎 = 𝐴𝑎 = 𝐴)
18 2fveq3 6893 . . . 4 (𝑎 = 𝐴 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄𝐴)))
1917, 18opeq12d 4880 . . 3 (𝑎 = 𝐴 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
2016, 19eqeq12d 2749 . 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 6889 . . . 4 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
23 opex 5463 . . . . 5 ⟨∅, ( I ‘𝐼)⟩ ∈ V
2423rdg0 8416 . . . 4 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
2522, 24eqtri 2761 . . 3 (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩
26 0ex 5306 . . . . . . 7 ∅ ∈ V
27 fvex 6901 . . . . . . 7 ( I ‘𝐼) ∈ V
2826, 27op2nd 7979 . . . . . 6 (2nd ‘⟨∅, ( I ‘𝐼)⟩) = ( I ‘𝐼)
2928eqcomi 2742 . . . . 5 ( I ‘𝐼) = (2nd ‘⟨∅, ( I ‘𝐼)⟩)
3029opeq2i 4876 . . . 4 ⟨∅, ( I ‘𝐼)⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩
31 id 22 . . . 4 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩)
32 fveq2 6888 . . . . 5 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (2nd ‘(𝑄‘∅)) = (2nd ‘⟨∅, ( I ‘𝐼)⟩))
3332opeq2d 4879 . . . 4 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → ⟨∅, (2nd ‘(𝑄‘∅))⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩)
3430, 31, 333eqtr4a 2799 . . 3 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
3525, 34ax-mp 5 . 2 (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩
36 df-ov 7407 . . . . . 6 (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
37 fvex 6901 . . . . . . 7 (2nd ‘(𝑄𝑏)) ∈ V
38 suceq 6427 . . . . . . . . 9 (𝑖 = 𝑏 → suc 𝑖 = suc 𝑏)
39 oveq1 7411 . . . . . . . . 9 (𝑖 = 𝑏 → (𝑖𝐹𝑣) = (𝑏𝐹𝑣))
4038, 39opeq12d 4880 . . . . . . . 8 (𝑖 = 𝑏 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹𝑣)⟩)
41 oveq2 7412 . . . . . . . . 9 (𝑣 = (2nd ‘(𝑄𝑏)) → (𝑏𝐹𝑣) = (𝑏𝐹(2nd ‘(𝑄𝑏))))
4241opeq2d 4879 . . . . . . . 8 (𝑣 = (2nd ‘(𝑄𝑏)) → ⟨suc 𝑏, (𝑏𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
43 eqid 2733 . . . . . . . 8 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
44 opex 5463 . . . . . . . 8 ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ ∈ V
4540, 42, 43, 44ovmpo 7563 . . . . . . 7 ((𝑏 ∈ ω ∧ (2nd ‘(𝑄𝑏)) ∈ V) → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
4637, 45mpan2 690 . . . . . 6 (𝑏 ∈ ω → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
4736, 46eqtr3id 2787 . . . . 5 (𝑏 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
48 fveqeq2 6897 . . . . 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 246 . . . 4 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩))
50 vex 3479 . . . . . . . . . 10 𝑏 ∈ V
5150sucex 7789 . . . . . . . . 9 suc 𝑏 ∈ V
52 ovex 7437 . . . . . . . . 9 (𝑏𝐹(2nd ‘(𝑄𝑏))) ∈ V
5351, 52op2nd 7979 . . . . . . . 8 (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩) = (𝑏𝐹(2nd ‘(𝑄𝑏)))
5453eqcomi 2742 . . . . . . 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 4879 . . . . 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 6888 . . . . . . 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 4879 . . . . . 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 2749 . . . . 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 246 . . . 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 8432 . . . . 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 7876 . . . . . . 7 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
6564fvresd 6908 . . . . . 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 6889 . . . . . 6 (𝑄‘suc 𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝑏)
6765, 66eqtr4di 2791 . . . . 5 (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝑏) = (𝑄‘suc 𝑏))
68 fvres 6907 . . . . . . 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 6889 . . . . . . 7 (𝑄𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝑏)
7068, 69eqtr4di 2791 . . . . . 6 (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝑏) = (𝑄𝑏))
7170fveq2d 6892 . . . . 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 2781 . . . 4 (𝑏 ∈ ω → (𝑄‘suc 𝑏) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))
7372fveq2d 6892 . . . . 5 (𝑏 ∈ ω → (2nd ‘(𝑄‘suc 𝑏)) = (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏))))
7473opeq2d 4879 . . . 4 (𝑏 ∈ ω → ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩ = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩)
7572, 74eqeq12d 2749 . . 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 259 . 2 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
775, 10, 15, 20, 35, 76finds 7884 1 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  c0 4321  cop 4633   I cid 5572  cres 5677  suc csuc 6363  cfv 6540  (class class class)co 7404  cmpo 7406  ωcom 7850  2nd c2nd 7969  reccrdg 8404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405
This theorem is referenced by:  seqomlem2  8446  seqomlem4  8448
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