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Theorem uzrdglem 13318
 Description: A helper lemma for the value of a recursive definition generator on upper integers. (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
uzrdglem (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
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 uzrdglem
StepHypRef Expression
1 om2uz.1 . . . . . 6 𝐶 ∈ ℤ
2 om2uz.2 . . . . . 6 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
31, 2om2uzf1oi 13314 . . . . 5 𝐺:ω–1-1-onto→(ℤ𝐶)
4 f1ocnvdm 7038 . . . . 5 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺𝐵) ∈ ω)
53, 4mpan 686 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝐺𝐵) ∈ ω)
6 uzrdg.1 . . . . 5 𝐴 ∈ V
7 uzrdg.2 . . . . 5 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
81, 2, 6, 7om2uzrdg 13317 . . . 4 ((𝐺𝐵) ∈ ω → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
95, 8syl 17 . . 3 (𝐵 ∈ (ℤ𝐶) → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
10 f1ocnvfv2 7031 . . . . 5 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐺𝐵)) = 𝐵)
113, 10mpan 686 . . . 4 (𝐵 ∈ (ℤ𝐶) → (𝐺‘(𝐺𝐵)) = 𝐵)
1211opeq1d 4807 . . 3 (𝐵 ∈ (ℤ𝐶) → ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩ = ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
139, 12eqtrd 2860 . 2 (𝐵 ∈ (ℤ𝐶) → (𝑅‘(𝐺𝐵)) = ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
14 frfnom 8064 . . . 4 (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
157fneq1i 6446 . . . 4 (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω)
1614, 15mpbir 232 . . 3 𝑅 Fn ω
17 fnfvelrn 6843 . . 3 ((𝑅 Fn ω ∧ (𝐺𝐵) ∈ ω) → (𝑅‘(𝐺𝐵)) ∈ ran 𝑅)
1816, 5, 17sylancr 587 . 2 (𝐵 ∈ (ℤ𝐶) → (𝑅‘(𝐺𝐵)) ∈ ran 𝑅)
1913, 18eqeltrrd 2918 1 (𝐵 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ∈ wcel 2106  Vcvv 3499  ⟨cop 4569   ↦ cmpt 5142  ◡ccnv 5552  ran crn 5554   ↾ cres 5555   Fn wfn 6346  –1-1-onto→wf1o 6350  ‘cfv 6351  (class class class)co 7151   ∈ cmpo 7153  ωcom 7571  2nd c2nd 7682  reccrdg 8039  1c1 10530   + caddc 10532  ℤcz 11973  ℤ≥cuz 12235 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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236 This theorem is referenced by:  uzrdgfni  13319  uzrdgsuci  13321
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