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Theorem om2uzrdg 13957
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 13948. (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 6896 . . 3 (𝑧 = ∅ → (𝑅𝑧) = (𝑅‘∅))
2 fveq2 6896 . . . 4 (𝑧 = ∅ → (𝐺𝑧) = (𝐺‘∅))
3 2fveq3 6901 . . . 4 (𝑧 = ∅ → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘∅)))
42, 3opeq12d 4883 . . 3 (𝑧 = ∅ → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
51, 4eqeq12d 2741 . 2 (𝑧 = ∅ → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
6 fveq2 6896 . . 3 (𝑧 = 𝑣 → (𝑅𝑧) = (𝑅𝑣))
7 fveq2 6896 . . . 4 (𝑧 = 𝑣 → (𝐺𝑧) = (𝐺𝑣))
8 2fveq3 6901 . . . 4 (𝑧 = 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝑣)))
97, 8opeq12d 4883 . . 3 (𝑧 = 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
106, 9eqeq12d 2741 . 2 (𝑧 = 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
11 fveq2 6896 . . 3 (𝑧 = suc 𝑣 → (𝑅𝑧) = (𝑅‘suc 𝑣))
12 fveq2 6896 . . . 4 (𝑧 = suc 𝑣 → (𝐺𝑧) = (𝐺‘suc 𝑣))
13 2fveq3 6901 . . . 4 (𝑧 = suc 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
1412, 13opeq12d 4883 . . 3 (𝑧 = suc 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
1511, 14eqeq12d 2741 . 2 (𝑧 = suc 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
16 fveq2 6896 . . 3 (𝑧 = 𝐵 → (𝑅𝑧) = (𝑅𝐵))
17 fveq2 6896 . . . 4 (𝑧 = 𝐵 → (𝐺𝑧) = (𝐺𝐵))
18 2fveq3 6901 . . . 4 (𝑧 = 𝐵 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝐵)))
1917, 18opeq12d 4883 . . 3 (𝑧 = 𝐵 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
2016, 19eqeq12d 2741 . 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 6897 . . . 4 (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅)
23 opex 5466 . . . . 5 𝐶, 𝐴⟩ ∈ V
24 fr0g 8457 . . . . 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 2753 . . 3 (𝑅‘∅) = ⟨𝐶, 𝐴
27 om2uz.1 . . . . 5 𝐶 ∈ ℤ
28 om2uz.2 . . . . 5 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
2927, 28om2uz0i 13948 . . . 4 (𝐺‘∅) = 𝐶
3026fveq2i 6899 . . . . 5 (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩)
3127elexi 3482 . . . . . 6 𝐶 ∈ V
32 uzrdg.1 . . . . . 6 𝐴 ∈ V
3331, 32op2nd 8003 . . . . 5 (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴
3430, 33eqtri 2753 . . . 4 (2nd ‘(𝑅‘∅)) = 𝐴
3529, 34opeq12i 4880 . . 3 ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴
3626, 35eqtr4i 2756 . 2 (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩
37 frsuc 8458 . . . . . 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 6897 . . . . . 6 (𝑅‘suc 𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣)
3921fveq1i 6897 . . . . . . 7 (𝑅𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)
4039fveq2i 6899 . . . . . 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 2790 . . . . 5 (𝑣 ∈ ω → (𝑅‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
42 fveq2 6896 . . . . . 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 7422 . . . . . . 7 ((𝐺𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
44 fvex 6909 . . . . . . . 8 (𝐺𝑣) ∈ V
45 fvex 6909 . . . . . . . 8 (2nd ‘(𝑅𝑣)) ∈ V
46 oveq1 7426 . . . . . . . . . 10 (𝑤 = (𝐺𝑣) → (𝑤 + 1) = ((𝐺𝑣) + 1))
47 oveq1 7426 . . . . . . . . . 10 (𝑤 = (𝐺𝑣) → (𝑤𝐹𝑧) = ((𝐺𝑣)𝐹𝑧))
4846, 47opeq12d 4883 . . . . . . . . 9 (𝑤 = (𝐺𝑣) → ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩)
49 oveq2 7427 . . . . . . . . . 10 (𝑧 = (2nd ‘(𝑅𝑣)) → ((𝐺𝑣)𝐹𝑧) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
5049opeq2d 4882 . . . . . . . . 9 (𝑧 = (2nd ‘(𝑅𝑣)) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
51 oveq1 7426 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑥 + 1) = (𝑤 + 1))
52 oveq1 7426 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
5351, 52opeq12d 4883 . . . . . . . . . 10 (𝑥 = 𝑤 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩)
54 oveq2 7427 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
5554opeq2d 4882 . . . . . . . . . 10 (𝑦 = 𝑧 → ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
5653, 55cbvmpov 7515 . . . . . . . . 9 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
57 opex 5466 . . . . . . . . 9 ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ V
5848, 50, 56, 57ovmpo 7581 . . . . . . . 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 690 . . . . . . 7 ((𝐺𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩
6043, 59eqtr3i 2755 . . . . . 6 ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩
6142, 60eqtrdi 2781 . . . . 5 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
6241, 61sylan9eq 2785 . . . 4 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
6327, 28om2uzsuci 13949 . . . . . 6 (𝑣 ∈ ω → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
6463adantr 479 . . . . 5 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
6562fveq2d 6900 . . . . . 6 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩))
66 ovex 7452 . . . . . . 7 ((𝐺𝑣) + 1) ∈ V
67 ovex 7452 . . . . . . 7 ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ V
6866, 67op2nd 8003 . . . . . 6 (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))
6965, 68eqtrdi 2781 . . . . 5 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
7064, 69opeq12d 4883 . . . 4 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
7162, 70eqtr4d 2768 . . 3 ((𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
7271ex 411 . 2 (𝑣 ∈ ω → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
735, 10, 15, 20, 36, 72finds 7904 1 (𝐵 ∈ ω → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  c0 4322  cop 4636  cmpt 5232  cres 5680  suc csuc 6373  cfv 6549  (class class class)co 7419  cmpo 7421  ωcom 7871  2nd c2nd 7993  reccrdg 8430  1c1 11141   + caddc 11143  cz 12591
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431
This theorem is referenced by:  uzrdglem  13958  uzrdgfni  13959  uzrdgsuci  13961
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