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Theorem uzrdgsuci 13922
Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13918. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
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)
uzrdg.3 𝑆 = ran 𝑅
Assertion
Ref Expression
uzrdgsuci (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆𝐵)))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐶   𝑦,𝐺   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem uzrdgsuci
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om2uz.1 . . . . . 6 𝐶 ∈ ℤ
2 om2uz.2 . . . . . 6 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
3 uzrdg.1 . . . . . 6 𝐴 ∈ V
4 uzrdg.2 . . . . . 6 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
5 uzrdg.3 . . . . . 6 𝑆 = ran 𝑅
61, 2, 3, 4, 5uzrdgfni 13920 . . . . 5 𝑆 Fn (ℤ𝐶)
7 fnfun 6639 . . . . 5 (𝑆 Fn (ℤ𝐶) → Fun 𝑆)
86, 7ax-mp 5 . . . 4 Fun 𝑆
9 peano2uz 12882 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → (𝐵 + 1) ∈ (ℤ𝐶))
101, 2, 3, 4uzrdglem 13919 . . . . . 6 ((𝐵 + 1) ∈ (ℤ𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ ran 𝑅)
119, 10syl 17 . . . . 5 (𝐵 ∈ (ℤ𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ ran 𝑅)
1211, 5eleqtrrdi 2836 . . . 4 (𝐵 ∈ (ℤ𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆)
13 funopfv 6933 . . . 4 (Fun 𝑆 → (⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆 → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))))
148, 12, 13mpsyl 68 . . 3 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1)))))
151, 2om2uzf1oi 13915 . . . . . . . 8 𝐺:ω–1-1-onto→(ℤ𝐶)
16 f1ocnvdm 7275 . . . . . . . 8 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺𝐵) ∈ ω)
1715, 16mpan 687 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐺𝐵) ∈ ω)
18 peano2 7874 . . . . . . 7 ((𝐺𝐵) ∈ ω → suc (𝐺𝐵) ∈ ω)
1917, 18syl 17 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → suc (𝐺𝐵) ∈ ω)
201, 2om2uzsuci 13910 . . . . . . . 8 ((𝐺𝐵) ∈ ω → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
2117, 20syl 17 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
22 f1ocnvfv2 7267 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐺𝐵)) = 𝐵)
2315, 22mpan 687 . . . . . . . 8 (𝐵 ∈ (ℤ𝐶) → (𝐺‘(𝐺𝐵)) = 𝐵)
2423oveq1d 7416 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → ((𝐺‘(𝐺𝐵)) + 1) = (𝐵 + 1))
2521, 24eqtrd 2764 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → (𝐺‘suc (𝐺𝐵)) = (𝐵 + 1))
26 f1ocnvfv 7268 . . . . . . 7 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ suc (𝐺𝐵) ∈ ω) → ((𝐺‘suc (𝐺𝐵)) = (𝐵 + 1) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵)))
2715, 26mpan 687 . . . . . 6 (suc (𝐺𝐵) ∈ ω → ((𝐺‘suc (𝐺𝐵)) = (𝐵 + 1) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵)))
2819, 25, 27sylc 65 . . . . 5 (𝐵 ∈ (ℤ𝐶) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵))
2928fveq2d 6885 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝑅‘(𝐺‘(𝐵 + 1))) = (𝑅‘suc (𝐺𝐵)))
3029fveq2d 6885 . . 3 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
3114, 30eqtrd 2764 . 2 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
32 frsuc 8432 . . . . . . . 8 ((𝐺𝐵) ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
334fveq1i 6882 . . . . . . . 8 (𝑅‘suc (𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵))
344fveq1i 6882 . . . . . . . . 9 (𝑅‘(𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))
3534fveq2i 6884 . . . . . . . 8 ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵)))
3632, 33, 353eqtr4g 2789 . . . . . . 7 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))))
371, 2, 3, 4om2uzrdg 13918 . . . . . . . . 9 ((𝐺𝐵) ∈ ω → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
3837fveq2d 6885 . . . . . . . 8 ((𝐺𝐵) ∈ ω → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩))
39 df-ov 7404 . . . . . . . 8 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
4038, 39eqtr4di 2782 . . . . . . 7 ((𝐺𝐵) ∈ ω → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
4136, 40eqtrd 2764 . . . . . 6 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
42 fvex 6894 . . . . . . 7 (𝐺‘(𝐺𝐵)) ∈ V
43 fvex 6894 . . . . . . 7 (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V
44 oveq1 7408 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧 + 1) = ((𝐺‘(𝐺𝐵)) + 1))
45 oveq1 7408 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹𝑤))
4644, 45opeq12d 4873 . . . . . . . 8 (𝑧 = (𝐺‘(𝐺𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩)
47 oveq2 7409 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ((𝐺‘(𝐺𝐵))𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
4847opeq2d 4872 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
49 oveq1 7408 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
50 oveq1 7408 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
5149, 50opeq12d 4873 . . . . . . . . 9 (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
52 oveq2 7409 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
5352opeq2d 4872 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
5451, 53cbvmpov 7496 . . . . . . . 8 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
55 opex 5454 . . . . . . . 8 ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ V
5646, 48, 54, 55ovmpo 7560 . . . . . . 7 (((𝐺‘(𝐺𝐵)) ∈ V ∧ (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V) → ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
5742, 43, 56mp2an 689 . . . . . 6 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩
5841, 57eqtrdi 2780 . . . . 5 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
5958fveq2d 6885 . . . 4 ((𝐺𝐵) ∈ ω → (2nd ‘(𝑅‘suc (𝐺𝐵))) = (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩))
60 ovex 7434 . . . . 5 ((𝐺‘(𝐺𝐵)) + 1) ∈ V
61 ovex 7434 . . . . 5 ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ V
6260, 61op2nd 7977 . . . 4 (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))
6359, 62eqtrdi 2780 . . 3 ((𝐺𝐵) ∈ ω → (2nd ‘(𝑅‘suc (𝐺𝐵))) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
6417, 63syl 17 . 2 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
651, 2, 3, 4uzrdglem 13919 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
6665, 5eleqtrrdi 2836 . . . . 5 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆)
67 funopfv 6933 . . . . 5 (Fun 𝑆 → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆 → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
688, 66, 67mpsyl 68 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵))))
6968eqcomd 2730 . . 3 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘(𝐺𝐵))) = (𝑆𝐵))
7023, 69oveq12d 7419 . 2 (𝐵 ∈ (ℤ𝐶) → ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) = (𝐵𝐹(𝑆𝐵)))
7131, 64, 703eqtrd 2768 1 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3466  cop 4626  cmpt 5221  ccnv 5665  ran crn 5667  cres 5668  suc csuc 6356  Fun wfun 6527   Fn wfn 6528  1-1-ontowf1o 6532  cfv 6533  (class class class)co 7401  cmpo 7403  ωcom 7848  2nd c2nd 7967  reccrdg 8404  1c1 11107   + caddc 11109  cz 12555  cuz 12819
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820
This theorem is referenced by:  seqp1  13978
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