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Theorem uzrdgsuci 13382
 Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13378. (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 13380 . . . . 5 𝑆 Fn (ℤ𝐶)
7 fnfun 6438 . . . . 5 (𝑆 Fn (ℤ𝐶) → Fun 𝑆)
86, 7ax-mp 5 . . . 4 Fun 𝑆
9 peano2uz 12346 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → (𝐵 + 1) ∈ (ℤ𝐶))
101, 2, 3, 4uzrdglem 13379 . . . . . 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 2863 . . . 4 (𝐵 ∈ (ℤ𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆)
13 funopfv 6709 . . . 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 13375 . . . . . . . 8 𝐺:ω–1-1-onto→(ℤ𝐶)
16 f1ocnvdm 7038 . . . . . . . 8 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺𝐵) ∈ ω)
1715, 16mpan 689 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐺𝐵) ∈ ω)
18 peano2 7606 . . . . . . 7 ((𝐺𝐵) ∈ ω → suc (𝐺𝐵) ∈ ω)
1917, 18syl 17 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → suc (𝐺𝐵) ∈ ω)
201, 2om2uzsuci 13370 . . . . . . . 8 ((𝐺𝐵) ∈ ω → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
2117, 20syl 17 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
22 f1ocnvfv2 7031 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐺𝐵)) = 𝐵)
2315, 22mpan 689 . . . . . . . 8 (𝐵 ∈ (ℤ𝐶) → (𝐺‘(𝐺𝐵)) = 𝐵)
2423oveq1d 7170 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → ((𝐺‘(𝐺𝐵)) + 1) = (𝐵 + 1))
2521, 24eqtrd 2793 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → (𝐺‘suc (𝐺𝐵)) = (𝐵 + 1))
26 f1ocnvfv 7032 . . . . . . 7 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ suc (𝐺𝐵) ∈ ω) → ((𝐺‘suc (𝐺𝐵)) = (𝐵 + 1) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵)))
2715, 26mpan 689 . . . . . 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 6666 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝑅‘(𝐺‘(𝐵 + 1))) = (𝑅‘suc (𝐺𝐵)))
3029fveq2d 6666 . . 3 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
3114, 30eqtrd 2793 . 2 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
32 frsuc 8087 . . . . . . . 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 6663 . . . . . . . 8 (𝑅‘suc (𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵))
344fveq1i 6663 . . . . . . . . 9 (𝑅‘(𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))
3534fveq2i 6665 . . . . . . . 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 2818 . . . . . . 7 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))))
371, 2, 3, 4om2uzrdg 13378 . . . . . . . . 9 ((𝐺𝐵) ∈ ω → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
3837fveq2d 6666 . . . . . . . 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 7158 . . . . . . . 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 2811 . . . . . . 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 2793 . . . . . 6 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
42 fvex 6675 . . . . . . 7 (𝐺‘(𝐺𝐵)) ∈ V
43 fvex 6675 . . . . . . 7 (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V
44 oveq1 7162 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧 + 1) = ((𝐺‘(𝐺𝐵)) + 1))
45 oveq1 7162 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹𝑤))
4644, 45opeq12d 4774 . . . . . . . 8 (𝑧 = (𝐺‘(𝐺𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩)
47 oveq2 7163 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ((𝐺‘(𝐺𝐵))𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
4847opeq2d 4773 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
49 oveq1 7162 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
50 oveq1 7162 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
5149, 50opeq12d 4774 . . . . . . . . 9 (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
52 oveq2 7163 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
5352opeq2d 4773 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
5451, 53cbvmpov 7248 . . . . . . . 8 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
55 opex 5327 . . . . . . . 8 ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ V
5646, 48, 54, 55ovmpo 7310 . . . . . . 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 691 . . . . . 6 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩
5841, 57eqtrdi 2809 . . . . 5 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
5958fveq2d 6666 . . . 4 ((𝐺𝐵) ∈ ω → (2nd ‘(𝑅‘suc (𝐺𝐵))) = (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩))
60 ovex 7188 . . . . 5 ((𝐺‘(𝐺𝐵)) + 1) ∈ V
61 ovex 7188 . . . . 5 ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ V
6260, 61op2nd 7707 . . . 4 (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))
6359, 62eqtrdi 2809 . . 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 13379 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
6665, 5eleqtrrdi 2863 . . . . 5 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆)
67 funopfv 6709 . . . . 5 (Fun 𝑆 → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆 → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
688, 66, 67mpsyl 68 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵))))
6968eqcomd 2764 . . 3 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘(𝐺𝐵))) = (𝑆𝐵))
7023, 69oveq12d 7173 . 2 (𝐵 ∈ (ℤ𝐶) → ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) = (𝐵𝐹(𝑆𝐵)))
7131, 64, 703eqtrd 2797 1 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4531   ↦ cmpt 5115  ◡ccnv 5526  ran crn 5528   ↾ cres 5529  suc csuc 6175  Fun wfun 6333   Fn wfn 6334  –1-1-onto→wf1o 6338  ‘cfv 6339  (class class class)co 7155   ∈ cmpo 7157  ωcom 7584  2nd c2nd 7697  reccrdg 8060  1c1 10581   + caddc 10583  ℤcz 12025  ℤ≥cuz 12287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-n0 11940  df-z 12026  df-uz 12288 This theorem is referenced by:  seqp1  13438
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