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Theorem seqomlem1 7590
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 6229 . . 3 (𝑎 = ∅ → (𝑄𝑎) = (𝑄‘∅))
2 id 22 . . . 4 (𝑎 = ∅ → 𝑎 = ∅)
31fveq2d 6233 . . . 4 (𝑎 = ∅ → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄‘∅)))
42, 3opeq12d 4441 . . 3 (𝑎 = ∅ → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
51, 4eqeq12d 2666 . 2 (𝑎 = ∅ → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩))
6 fveq2 6229 . . 3 (𝑎 = 𝑏 → (𝑄𝑎) = (𝑄𝑏))
7 id 22 . . . 4 (𝑎 = 𝑏𝑎 = 𝑏)
86fveq2d 6233 . . . 4 (𝑎 = 𝑏 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄𝑏)))
97, 8opeq12d 4441 . . 3 (𝑎 = 𝑏 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
106, 9eqeq12d 2666 . 2 (𝑎 = 𝑏 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩))
11 fveq2 6229 . . 3 (𝑎 = suc 𝑏 → (𝑄𝑎) = (𝑄‘suc 𝑏))
12 id 22 . . . 4 (𝑎 = suc 𝑏𝑎 = suc 𝑏)
1311fveq2d 6233 . . . 4 (𝑎 = suc 𝑏 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄‘suc 𝑏)))
1412, 13opeq12d 4441 . . 3 (𝑎 = suc 𝑏 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩)
1511, 14eqeq12d 2666 . 2 (𝑎 = suc 𝑏 → ((𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
16 fveq2 6229 . . 3 (𝑎 = 𝐴 → (𝑄𝑎) = (𝑄𝐴))
17 id 22 . . . 4 (𝑎 = 𝐴𝑎 = 𝐴)
1816fveq2d 6233 . . . 4 (𝑎 = 𝐴 → (2nd ‘(𝑄𝑎)) = (2nd ‘(𝑄𝐴)))
1917, 18opeq12d 4441 . . 3 (𝑎 = 𝐴 → ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
2016, 19eqeq12d 2666 . 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 6230 . . . 4 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
23 opex 4962 . . . . 5 ⟨∅, ( I ‘𝐼)⟩ ∈ V
2423rdg0 7562 . . . 4 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
2522, 24eqtri 2673 . . 3 (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩
26 0ex 4823 . . . . . . 7 ∅ ∈ V
27 fvex 6239 . . . . . . 7 ( I ‘𝐼) ∈ V
2826, 27op2nd 7219 . . . . . 6 (2nd ‘⟨∅, ( I ‘𝐼)⟩) = ( I ‘𝐼)
2928eqcomi 2660 . . . . 5 ( I ‘𝐼) = (2nd ‘⟨∅, ( I ‘𝐼)⟩)
3029opeq2i 4437 . . . 4 ⟨∅, ( I ‘𝐼)⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩
31 id 22 . . . 4 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩)
32 fveq2 6229 . . . . 5 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (2nd ‘(𝑄‘∅)) = (2nd ‘⟨∅, ( I ‘𝐼)⟩))
3332opeq2d 4440 . . . 4 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → ⟨∅, (2nd ‘(𝑄‘∅))⟩ = ⟨∅, (2nd ‘⟨∅, ( I ‘𝐼)⟩)⟩)
3430, 31, 333eqtr4a 2711 . . 3 ((𝑄‘∅) = ⟨∅, ( I ‘𝐼)⟩ → (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩)
3525, 34ax-mp 5 . 2 (𝑄‘∅) = ⟨∅, (2nd ‘(𝑄‘∅))⟩
36 df-ov 6693 . . . . . 6 (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
37 fvex 6239 . . . . . . 7 (2nd ‘(𝑄𝑏)) ∈ V
38 suceq 5828 . . . . . . . . 9 (𝑖 = 𝑏 → suc 𝑖 = suc 𝑏)
39 oveq1 6697 . . . . . . . . 9 (𝑖 = 𝑏 → (𝑖𝐹𝑣) = (𝑏𝐹𝑣))
4038, 39opeq12d 4441 . . . . . . . 8 (𝑖 = 𝑏 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹𝑣)⟩)
41 oveq2 6698 . . . . . . . . 9 (𝑣 = (2nd ‘(𝑄𝑏)) → (𝑏𝐹𝑣) = (𝑏𝐹(2nd ‘(𝑄𝑏))))
4241opeq2d 4440 . . . . . . . 8 (𝑣 = (2nd ‘(𝑄𝑏)) → ⟨suc 𝑏, (𝑏𝐹𝑣)⟩ = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
43 eqid 2651 . . . . . . . 8 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
44 opex 4962 . . . . . . . 8 ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ ∈ V
4540, 42, 43, 44ovmpt2 6838 . . . . . . 7 ((𝑏 ∈ ω ∧ (2nd ‘(𝑄𝑏)) ∈ V) → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
4637, 45mpan2 707 . . . . . 6 (𝑏 ∈ ω → (𝑏(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝑏))) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
4736, 46syl5eqr 2699 . . . . 5 (𝑏 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
48 fveq2 6229 . . . . . 6 ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝑏, (2nd ‘(𝑄𝑏))⟩))
4948eqeq1d 2653 . . . . 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)))⟩))
5047, 49syl5ibrcom 237 . . . 4 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)) = ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩))
51 vex 3234 . . . . . . . . . 10 𝑏 ∈ V
5251sucex 7053 . . . . . . . . 9 suc 𝑏 ∈ V
53 ovex 6718 . . . . . . . . 9 (𝑏𝐹(2nd ‘(𝑄𝑏))) ∈ V
5452, 53op2nd 7219 . . . . . . . 8 (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩) = (𝑏𝐹(2nd ‘(𝑄𝑏)))
5554eqcomi 2660 . . . . . . 7 (𝑏𝐹(2nd ‘(𝑄𝑏))) = (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)
5655a1i 11 . . . . . 6 (𝑏 ∈ ω → (𝑏𝐹(2nd ‘(𝑄𝑏))) = (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩))
5756opeq2d 4440 . . . . 5 (𝑏 ∈ ω → ⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩ = ⟨suc 𝑏, (2nd ‘⟨suc 𝑏, (𝑏𝐹(2nd ‘(𝑄𝑏)))⟩)⟩)
58 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)))⟩)
59 fveq2 6229 . . . . . . 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)))⟩))
6059opeq2d 4440 . . . . . 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)))⟩)⟩)
6158, 60eqeq12d 2666 . . . . 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)))⟩)⟩))
6257, 61syl5ibrcom 237 . . . 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)))⟩))
6350, 62syld 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)))⟩))
64 frsuc 7577 . . . . 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)))
65 peano2 7128 . . . . . . 7 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
66 fvres 6245 . . . . . . 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))
6765, 66syl 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))
6821fveq1i 6230 . . . . . 6 (𝑄‘suc 𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝑏)
6967, 68syl6eqr 2703 . . . . 5 (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝑏) = (𝑄‘suc 𝑏))
70 fvres 6245 . . . . . . 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))
7121fveq1i 6230 . . . . . . 7 (𝑄𝑏) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝑏)
7270, 71syl6eqr 2703 . . . . . 6 (𝑏 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝑏) = (𝑄𝑏))
7372fveq2d 6233 . . . . 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)))
7464, 69, 733eqtr3d 2693 . . . 4 (𝑏 ∈ ω → (𝑄‘suc 𝑏) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))
7574fveq2d 6233 . . . . 5 (𝑏 ∈ ω → (2nd ‘(𝑄‘suc 𝑏)) = (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏))))
7675opeq2d 4440 . . . 4 (𝑏 ∈ ω → ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩ = ⟨suc 𝑏, (2nd ‘((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝑏)))⟩)
7774, 76eqeq12d 2666 . . 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)))⟩))
7863, 77sylibrd 249 . 2 (𝑏 ∈ ω → ((𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ → (𝑄‘suc 𝑏) = ⟨suc 𝑏, (2nd ‘(𝑄‘suc 𝑏))⟩))
795, 10, 15, 20, 35, 78finds 7134 1 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  cop 4216   I cid 5052  cres 5145  suc csuc 5763  cfv 5926  (class class class)co 6690  cmpt2 6692  ωcom 7107  2nd c2nd 7209  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  seqomlem2  7591  seqomlem4  7593
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