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Theorem seqomlem1 7753
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 6379 . . 3 (𝑎 = ∅ → (𝑄𝑎) = (𝑄‘∅))
2 id 22 . . . 4 (𝑎 = ∅ → 𝑎 = ∅)
3 2fveq3 6384 . . . 4 (𝑎 = ∅ → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄‘∅)))
42, 3opeq12d 4569 . . 3 (𝑎 = ∅ → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
51, 4eqeq12d 2780 . 2 (𝑎 = ∅ → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩))
6 fveq2 6379 . . 3 (𝑎 = 𝑏 → (𝑄𝑎) = (𝑄𝑏))
7 id 22 . . . 4 (𝑎 = 𝑏𝑎 = 𝑏)
8 2fveq3 6384 . . . 4 (𝑎 = 𝑏 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄𝑏)))
97, 8opeq12d 4569 . . 3 (𝑎 = 𝑏 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
106, 9eqeq12d 2780 . 2 (𝑎 = 𝑏 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩))
11 fveq2 6379 . . 3 (𝑎 = suc 𝑏 → (𝑄𝑎) = (𝑄‘suc 𝑏))
12 id 22 . . . 4 (𝑎 = suc 𝑏𝑎 = suc 𝑏)
13 2fveq3 6384 . . . 4 (𝑎 = suc 𝑏 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄‘suc 𝑏)))
1412, 13opeq12d 4569 . . 3 (𝑎 = suc 𝑏 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩)
1511, 14eqeq12d 2780 . 2 (𝑎 = suc 𝑏 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
16 fveq2 6379 . . 3 (𝑎 = 𝐴 → (𝑄𝑎) = (𝑄𝐴))
17 id 22 . . . 4 (𝑎 = 𝐴𝑎 = 𝐴)
18 2fveq3 6384 . . . 4 (𝑎 = 𝐴 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄𝐴)))
1917, 18opeq12d 4569 . . 3 (𝑎 = 𝐴 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
2016, 19eqeq12d 2780 . 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 6380 . . . 4 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
23 opex 5090 . . . . 5 ⟨∅, ( I ‘𝐼)⟩ ∈ V
2423rdg0 7725 . . . 4 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
2522, 24eqtri 2787 . . 3 (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩
26 0ex 4952 . . . . . . 7 ∅ ∈ V
27 fvex 6392 . . . . . . 7 ( I ‘𝐼) ∈ V
2826, 27op2nd 7379 . . . . . 6 (2nd ‘⟨∅, ( I ‘𝐼)⟩) = ( I ‘𝐼)
2928eqcomi 2774 . . . . 5 ( I ‘𝐼) = (2nd ‘⟨∅, ( I ‘𝐼)⟩)
3029opeq2i 4565 . . . 4 ⟨∅, ( I ‘𝐼)⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩
31 id 22 . . . 4 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩)
32 fveq2 6379 . . . . 5 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (2nd ‘(𝑄‘∅)) = (2nd ‘⟨∅, ( I ‘𝐼)⟩))
3332opeq2d 4568 . . . 4 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → ⟨∅, (2nd ‘(𝑄‘∅))⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩)
3430, 31, 333eqtr4a 2825 . . 3 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
3525, 34ax-mp 5 . 2 (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩
36 df-ov 6849 . . . . . 6 (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
37 fvex 6392 . . . . . . 7 (2nd ‘(𝑄𝑏)) ∈ V
38 suceq 5975 . . . . . . . . 9 (𝑖 = 𝑏 → suc 𝑖 = suc 𝑏)
39 oveq1 6853 . . . . . . . . 9 (𝑖 = 𝑏 → (𝑖𝐹𝑣) = (𝑏𝐹𝑣))
4038, 39opeq12d 4569 . . . . . . . 8 (𝑖 = 𝑏 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹𝑣)⟩)
41 oveq2 6854 . . . . . . . . 9 (𝑣 = (2nd ‘(𝑄𝑏)) → (𝑏𝐹𝑣) = (𝑏𝐹(2nd ‘(𝑄𝑏))))
4241opeq2d 4568 . . . . . . . 8 (𝑣 = (2nd ‘(𝑄𝑏)) → ⟨suc 𝑏, (𝑏𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
43 eqid 2765 . . . . . . . 8 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
44 opex 5090 . . . . . . . 8 ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ ∈ V
4540, 42, 43, 44ovmpt2 6998 . . . . . . 7 ((𝑏 ∈ ω ∧ (2nd ‘(𝑄𝑏)) ∈ V) → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
4637, 45mpan2 682 . . . . . 6 (𝑏 ∈ ω → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
4736, 46syl5eqr 2813 . . . . 5 (𝑏 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
48 fveqeq2 6388 . . . . 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 238 . . . 4 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩))
50 vex 3353 . . . . . . . . . 10 𝑏 ∈ V
5150sucex 7213 . . . . . . . . 9 suc 𝑏 ∈ V
52 ovex 6878 . . . . . . . . 9 (𝑏𝐹(2nd ‘(𝑄𝑏))) ∈ V
5351, 52op2nd 7379 . . . . . . . 8 (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩) = (𝑏𝐹(2nd ‘(𝑄𝑏)))
5453eqcomi 2774 . . . . . . 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 4568 . . . . 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 6379 . . . . . . 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 4568 . . . . . 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 2780 . . . . 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 238 . . . 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 7740 . . . . 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 7288 . . . . . . 7 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
65 fvres 6398 . . . . . . 7 (suc 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))
6664, 65syl 17 . . . . . 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))
6721fveq1i 6380 . . . . . 6 (𝑄‘suc 𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝑏)
6866, 67syl6eqr 2817 . . . . 5 (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝑏) = (𝑄‘suc 𝑏))
69 fvres 6398 . . . . . . 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))
7021fveq1i 6380 . . . . . . 7 (𝑄𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝑏)
7169, 70syl6eqr 2817 . . . . . 6 (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝑏) = (𝑄𝑏))
7271fveq2d 6383 . . . . 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)))
7363, 68, 723eqtr3d 2807 . . . 4 (𝑏 ∈ ω → (𝑄‘suc 𝑏) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))
7473fveq2d 6383 . . . . 5 (𝑏 ∈ ω → (2nd ‘(𝑄‘suc 𝑏)) = (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏))))
7574opeq2d 4568 . . . 4 (𝑏 ∈ ω → ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩ = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩)
7673, 75eqeq12d 2780 . . 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)))⟩))
7762, 76sylibrd 250 . 2 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
785, 10, 15, 20, 35, 77finds 7294 1 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  c0 4081  cop 4342   I cid 5186  cres 5281  suc csuc 5912  cfv 6070  (class class class)co 6846  cmpt2 6848  ωcom 7267  2nd c2nd 7369  reccrdg 7713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714
This theorem is referenced by:  seqomlem2  7754  seqomlem4  7756
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