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Theorem uzrdgsuci 12945
Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 12941. (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 12943 . . . . 5 𝑆 Fn (ℤ𝐶)
7 fnfun 6141 . . . . 5 (𝑆 Fn (ℤ𝐶) → Fun 𝑆)
86, 7ax-mp 5 . . . 4 Fun 𝑆
9 peano2uz 11926 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → (𝐵 + 1) ∈ (ℤ𝐶))
101, 2, 3, 4uzrdglem 12942 . . . . . 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, 5syl6eleqr 2842 . . . 4 (𝐵 ∈ (ℤ𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆)
13 funopfv 6388 . . . 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 12938 . . . . . . . 8 𝐺:ω–1-1-onto→(ℤ𝐶)
16 f1ocnvdm 6695 . . . . . . . 8 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺𝐵) ∈ ω)
1715, 16mpan 708 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐺𝐵) ∈ ω)
18 peano2 7243 . . . . . . 7 ((𝐺𝐵) ∈ ω → suc (𝐺𝐵) ∈ ω)
1917, 18syl 17 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → suc (𝐺𝐵) ∈ ω)
201, 2om2uzsuci 12933 . . . . . . . 8 ((𝐺𝐵) ∈ ω → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
2117, 20syl 17 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
22 f1ocnvfv2 6688 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐺𝐵)) = 𝐵)
2315, 22mpan 708 . . . . . . . 8 (𝐵 ∈ (ℤ𝐶) → (𝐺‘(𝐺𝐵)) = 𝐵)
2423oveq1d 6820 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → ((𝐺‘(𝐺𝐵)) + 1) = (𝐵 + 1))
2521, 24eqtrd 2786 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → (𝐺‘suc (𝐺𝐵)) = (𝐵 + 1))
26 f1ocnvfv 6689 . . . . . . 7 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ suc (𝐺𝐵) ∈ ω) → ((𝐺‘suc (𝐺𝐵)) = (𝐵 + 1) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵)))
2715, 26mpan 708 . . . . . 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 6348 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝑅‘(𝐺‘(𝐵 + 1))) = (𝑅‘suc (𝐺𝐵)))
3029fveq2d 6348 . . 3 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
3114, 30eqtrd 2786 . 2 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
32 frsuc 7693 . . . . . . . 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 6345 . . . . . . . 8 (𝑅‘suc (𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵))
344fveq1i 6345 . . . . . . . . 9 (𝑅‘(𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))
3534fveq2i 6347 . . . . . . . 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 2811 . . . . . . 7 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))))
371, 2, 3, 4om2uzrdg 12941 . . . . . . . . 9 ((𝐺𝐵) ∈ ω → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
3837fveq2d 6348 . . . . . . . 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 6808 . . . . . . . 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, 39syl6eqr 2804 . . . . . . 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 2786 . . . . . 6 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
42 fvex 6354 . . . . . . 7 (𝐺‘(𝐺𝐵)) ∈ V
43 fvex 6354 . . . . . . 7 (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V
44 oveq1 6812 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧 + 1) = ((𝐺‘(𝐺𝐵)) + 1))
45 oveq1 6812 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹𝑤))
4644, 45opeq12d 4553 . . . . . . . 8 (𝑧 = (𝐺‘(𝐺𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩)
47 oveq2 6813 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ((𝐺‘(𝐺𝐵))𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
4847opeq2d 4552 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
49 oveq1 6812 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
50 oveq1 6812 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
5149, 50opeq12d 4553 . . . . . . . . 9 (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
52 oveq2 6813 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
5352opeq2d 4552 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
5451, 53cbvmpt2v 6892 . . . . . . . 8 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
55 opex 5073 . . . . . . . 8 ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ V
5646, 48, 54, 55ovmpt2 6953 . . . . . . 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 710 . . . . . 6 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩
5841, 57syl6eq 2802 . . . . 5 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
5958fveq2d 6348 . . . 4 ((𝐺𝐵) ∈ ω → (2nd ‘(𝑅‘suc (𝐺𝐵))) = (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩))
60 ovex 6833 . . . . 5 ((𝐺‘(𝐺𝐵)) + 1) ∈ V
61 ovex 6833 . . . . 5 ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ V
6260, 61op2nd 7334 . . . 4 (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))
6359, 62syl6eq 2802 . . 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 12942 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
6665, 5syl6eleqr 2842 . . . . 5 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆)
67 funopfv 6388 . . . . 5 (Fun 𝑆 → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆 → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
688, 66, 67mpsyl 68 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵))))
6968eqcomd 2758 . . 3 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘(𝐺𝐵))) = (𝑆𝐵))
7023, 69oveq12d 6823 . 2 (𝐵 ∈ (ℤ𝐶) → ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) = (𝐵𝐹(𝑆𝐵)))
7131, 64, 703eqtrd 2790 1 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  wcel 2131  Vcvv 3332  cop 4319  cmpt 4873  ccnv 5257  ran crn 5259  cres 5260  suc csuc 5878  Fun wfun 6035   Fn wfn 6036  1-1-ontowf1o 6040  cfv 6041  (class class class)co 6805  cmpt2 6807  ωcom 7222  2nd c2nd 7324  reccrdg 7666  1c1 10121   + caddc 10123  cz 11561  cuz 11871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-n0 11477  df-z 11562  df-uz 11872
This theorem is referenced by:  seqp1  13002
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