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Theorem uzrdgsuci 14001
Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13997. (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 13999 . . . . 5 𝑆 Fn (ℤ𝐶)
7 fnfun 6668 . . . . 5 (𝑆 Fn (ℤ𝐶) → Fun 𝑆)
86, 7ax-mp 5 . . . 4 Fun 𝑆
9 peano2uz 12943 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → (𝐵 + 1) ∈ (ℤ𝐶))
101, 2, 3, 4uzrdglem 13998 . . . . . 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 2852 . . . 4 (𝐵 ∈ (ℤ𝐶) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑆)
13 funopfv 6958 . . . 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 13994 . . . . . . . 8 𝐺:ω–1-1-onto→(ℤ𝐶)
16 f1ocnvdm 7305 . . . . . . . 8 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺𝐵) ∈ ω)
1715, 16mpan 690 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐺𝐵) ∈ ω)
18 peano2 7912 . . . . . . 7 ((𝐺𝐵) ∈ ω → suc (𝐺𝐵) ∈ ω)
1917, 18syl 17 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → suc (𝐺𝐵) ∈ ω)
201, 2om2uzsuci 13989 . . . . . . . 8 ((𝐺𝐵) ∈ ω → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
2117, 20syl 17 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
22 f1ocnvfv2 7297 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐺𝐵)) = 𝐵)
2315, 22mpan 690 . . . . . . . 8 (𝐵 ∈ (ℤ𝐶) → (𝐺‘(𝐺𝐵)) = 𝐵)
2423oveq1d 7446 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → ((𝐺‘(𝐺𝐵)) + 1) = (𝐵 + 1))
2521, 24eqtrd 2777 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → (𝐺‘suc (𝐺𝐵)) = (𝐵 + 1))
26 f1ocnvfv 7298 . . . . . . 7 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ suc (𝐺𝐵) ∈ ω) → ((𝐺‘suc (𝐺𝐵)) = (𝐵 + 1) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵)))
2715, 26mpan 690 . . . . . 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 6910 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝑅‘(𝐺‘(𝐵 + 1))) = (𝑅‘suc (𝐺𝐵)))
3029fveq2d 6910 . . 3 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
3114, 30eqtrd 2777 . 2 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
32 frsuc 8477 . . . . . . . 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 6907 . . . . . . . 8 (𝑅‘suc (𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵))
344fveq1i 6907 . . . . . . . . 9 (𝑅‘(𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))
3534fveq2i 6909 . . . . . . . 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 2802 . . . . . . 7 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))))
371, 2, 3, 4om2uzrdg 13997 . . . . . . . . 9 ((𝐺𝐵) ∈ ω → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
3837fveq2d 6910 . . . . . . . 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 7434 . . . . . . . 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 2795 . . . . . . 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 2777 . . . . . 6 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
42 fvex 6919 . . . . . . 7 (𝐺‘(𝐺𝐵)) ∈ V
43 fvex 6919 . . . . . . 7 (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V
44 oveq1 7438 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧 + 1) = ((𝐺‘(𝐺𝐵)) + 1))
45 oveq1 7438 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹𝑤))
4644, 45opeq12d 4881 . . . . . . . 8 (𝑧 = (𝐺‘(𝐺𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩)
47 oveq2 7439 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ((𝐺‘(𝐺𝐵))𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
4847opeq2d 4880 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
49 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
50 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
5149, 50opeq12d 4881 . . . . . . . . 9 (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
52 oveq2 7439 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
5352opeq2d 4880 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
5451, 53cbvmpov 7528 . . . . . . . 8 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
55 opex 5469 . . . . . . . 8 ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ V
5646, 48, 54, 55ovmpo 7593 . . . . . . 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 692 . . . . . 6 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩
5841, 57eqtrdi 2793 . . . . 5 ((𝐺𝐵) ∈ ω → (𝑅‘suc (𝐺𝐵)) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
5958fveq2d 6910 . . . 4 ((𝐺𝐵) ∈ ω → (2nd ‘(𝑅‘suc (𝐺𝐵))) = (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩))
60 ovex 7464 . . . . 5 ((𝐺‘(𝐺𝐵)) + 1) ∈ V
61 ovex 7464 . . . . 5 ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ V
6260, 61op2nd 8023 . . . 4 (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))
6359, 62eqtrdi 2793 . . 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 13998 . . . . . 6 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
6665, 5eleqtrrdi 2852 . . . . 5 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆)
67 funopfv 6958 . . . . 5 (Fun 𝑆 → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆 → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
688, 66, 67mpsyl 68 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵))))
6968eqcomd 2743 . . 3 (𝐵 ∈ (ℤ𝐶) → (2nd ‘(𝑅‘(𝐺𝐵))) = (𝑆𝐵))
7023, 69oveq12d 7449 . 2 (𝐵 ∈ (ℤ𝐶) → ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) = (𝐵𝐹(𝑆𝐵)))
7131, 64, 703eqtrd 2781 1 (𝐵 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cmpt 5225  ccnv 5684  ran crn 5686  cres 5687  suc csuc 6386  Fun wfun 6555   Fn wfn 6556  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  2nd c2nd 8013  reccrdg 8449  1c1 11156   + caddc 11158  cz 12613  cuz 12878
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-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
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-nel 3047  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-riota 7388  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  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879
This theorem is referenced by:  seqp1  14057
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