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Theorem om2uzrdg 13997
Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either or 0) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. Normally 𝐹 is a function on the partition, and 𝐴 is a member of the partition. See also comment in om2uz0i 13988. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
om2uz.1 𝐶 ∈ ℤ
om2uz.2 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
uzrdg.1 𝐴 ∈ V
uzrdg.2 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
Assertion
Ref Expression
om2uzrdg (𝐵 ∈ ω → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐶   𝑦,𝐺   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem om2uzrdg
Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . 3 (𝑧 = ∅ → (𝑅𝑧) = (𝑅‘∅))
2 fveq2 6906 . . . 4 (𝑧 = ∅ → (𝐺𝑧) = (𝐺‘∅))
3 2fveq3 6911 . . . 4 (𝑧 = ∅ → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘∅)))
42, 3opeq12d 4881 . . 3 (𝑧 = ∅ → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
51, 4eqeq12d 2753 . 2 (𝑧 = ∅ → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
6 fveq2 6906 . . 3 (𝑧 = 𝑣 → (𝑅𝑧) = (𝑅𝑣))
7 fveq2 6906 . . . 4 (𝑧 = 𝑣 → (𝐺𝑧) = (𝐺𝑣))
8 2fveq3 6911 . . . 4 (𝑧 = 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝑣)))
97, 8opeq12d 4881 . . 3 (𝑧 = 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
106, 9eqeq12d 2753 . 2 (𝑧 = 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
11 fveq2 6906 . . 3 (𝑧 = suc 𝑣 → (𝑅𝑧) = (𝑅‘suc 𝑣))
12 fveq2 6906 . . . 4 (𝑧 = suc 𝑣 → (𝐺𝑧) = (𝐺‘suc 𝑣))
13 2fveq3 6911 . . . 4 (𝑧 = suc 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
1412, 13opeq12d 4881 . . 3 (𝑧 = suc 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
1511, 14eqeq12d 2753 . 2 (𝑧 = suc 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
16 fveq2 6906 . . 3 (𝑧 = 𝐵 → (𝑅𝑧) = (𝑅𝐵))
17 fveq2 6906 . . . 4 (𝑧 = 𝐵 → (𝐺𝑧) = (𝐺𝐵))
18 2fveq3 6911 . . . 4 (𝑧 = 𝐵 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝐵)))
1917, 18opeq12d 4881 . . 3 (𝑧 = 𝐵 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
2016, 19eqeq12d 2753 . 2 (𝑧 = 𝐵 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
21 uzrdg.2 . . . . 5 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
2221fveq1i 6907 . . . 4 (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅)
23 opex 5469 . . . . 5 𝐶, 𝐴⟩ ∈ V
24 fr0g 8476 . . . . 5 (⟨𝐶, 𝐴⟩ ∈ V → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩)
2523, 24ax-mp 5 . . . 4 ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴
2622, 25eqtri 2765 . . 3 (𝑅‘∅) = ⟨𝐶, 𝐴
27 om2uz.1 . . . . 5 𝐶 ∈ ℤ
28 om2uz.2 . . . . 5 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
2927, 28om2uz0i 13988 . . . 4 (𝐺‘∅) = 𝐶
3026fveq2i 6909 . . . . 5 (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩)
3127elexi 3503 . . . . . 6 𝐶 ∈ V
32 uzrdg.1 . . . . . 6 𝐴 ∈ V
3331, 32op2nd 8023 . . . . 5 (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴
3430, 33eqtri 2765 . . . 4 (2nd ‘(𝑅‘∅)) = 𝐴
3529, 34opeq12i 4878 . . 3 ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴
3626, 35eqtr4i 2768 . 2 (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩
37 frsuc 8477 . . . . . 6 (𝑣 ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
3821fveq1i 6907 . . . . . 6 (𝑅‘suc 𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣)
3921fveq1i 6907 . . . . . . 7 (𝑅𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)
4039fveq2i 6909 . . . . . 6 ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣))
4137, 38, 403eqtr4g 2802 . . . . 5 (𝑣 ∈ ω → (𝑅‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
42 fveq2 6906 . . . . . 6 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
43 df-ov 7434 . . . . . . 7 ((𝐺𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
44 fvex 6919 . . . . . . . 8 (𝐺𝑣) ∈ V
45 fvex 6919 . . . . . . . 8 (2nd ‘(𝑅𝑣)) ∈ V
46 oveq1 7438 . . . . . . . . . 10 (𝑤 = (𝐺𝑣) → (𝑤 + 1) = ((𝐺𝑣) + 1))
47 oveq1 7438 . . . . . . . . . 10 (𝑤 = (𝐺𝑣) → (𝑤𝐹𝑧) = ((𝐺𝑣)𝐹𝑧))
4846, 47opeq12d 4881 . . . . . . . . 9 (𝑤 = (𝐺𝑣) → ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩)
49 oveq2 7439 . . . . . . . . . 10 (𝑧 = (2nd ‘(𝑅𝑣)) → ((𝐺𝑣)𝐹𝑧) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
5049opeq2d 4880 . . . . . . . . 9 (𝑧 = (2nd ‘(𝑅𝑣)) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
51 oveq1 7438 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑥 + 1) = (𝑤 + 1))
52 oveq1 7438 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
5351, 52opeq12d 4881 . . . . . . . . . 10 (𝑥 = 𝑤 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩)
54 oveq2 7439 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
5554opeq2d 4880 . . . . . . . . . 10 (𝑦 = 𝑧 → ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
5653, 55cbvmpov 7528 . . . . . . . . 9 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
57 opex 5469 . . . . . . . . 9 ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ V
5848, 50, 56, 57ovmpo 7593 . . . . . . . 8 (((𝐺𝑣) ∈ V ∧ (2nd ‘(𝑅𝑣)) ∈ V) → ((𝐺𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
5944, 45, 58mp2an 692 . . . . . . 7 ((𝐺𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩
6043, 59eqtr3i 2767 . . . . . 6 ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩
6142, 60eqtrdi 2793 . . . . 5 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
6241, 61sylan9eq 2797 . . . 4 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
6327, 28om2uzsuci 13989 . . . . . 6 (𝑣 ∈ ω → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
6463adantr 480 . . . . 5 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
6562fveq2d 6910 . . . . . 6 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩))
66 ovex 7464 . . . . . . 7 ((𝐺𝑣) + 1) ∈ V
67 ovex 7464 . . . . . . 7 ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ V
6866, 67op2nd 8023 . . . . . 6 (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))
6965, 68eqtrdi 2793 . . . . 5 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
7064, 69opeq12d 4881 . . . 4 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
7162, 70eqtr4d 2780 . . 3 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
7271ex 412 . 2 (𝑣 ∈ ω → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
735, 10, 15, 20, 36, 72finds 7918 1 (𝐵 ∈ ω → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cop 4632  cmpt 5225  cres 5687  suc csuc 6386  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  2nd c2nd 8013  reccrdg 8449  1c1 11156   + caddc 11158  cz 12613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  uzrdglem  13998  uzrdgfni  13999  uzrdgsuci  14001
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