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Theorem frecuzrdgrrn 9194
 Description: The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. (Contributed by Jim Kingdon, 27-May-2020.)
Hypotheses
Ref Expression
frec2uz.1 (𝜑𝐶 ∈ ℤ)
frec2uz.2 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
uzrdg.s (𝜑𝑆𝑉)
uzrdg.a (𝜑𝐴𝑆)
uzrdg.f ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
uzrdg.2 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
Assertion
Ref Expression
frecuzrdgrrn ((𝜑𝐷 ∈ ω) → (𝑅𝐷) ∈ ((ℤ𝐶) × 𝑆))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐶,𝑦   𝑦,𝐺   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem frecuzrdgrrn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 uzrdg.2 . . 3 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
21fveq1i 5179 . 2 (𝑅𝐷) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝐷)
3 zex 8254 . . . . . . . 8 ℤ ∈ V
4 uzssz 8492 . . . . . . . 8 (ℤ𝐶) ⊆ ℤ
53, 4ssexi 3895 . . . . . . 7 (ℤ𝐶) ∈ V
65a1i 9 . . . . . 6 ((𝜑𝐷 ∈ ω) → (ℤ𝐶) ∈ V)
7 uzrdg.s . . . . . . 7 (𝜑𝑆𝑉)
87adantr 261 . . . . . 6 ((𝜑𝐷 ∈ ω) → 𝑆𝑉)
9 mpt2exga 5835 . . . . . 6 (((ℤ𝐶) ∈ V ∧ 𝑆𝑉) → (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) ∈ V)
106, 8, 9syl2anc 391 . . . . 5 ((𝜑𝐷 ∈ ω) → (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) ∈ V)
11 vex 2560 . . . . . 6 𝑧 ∈ V
1211a1i 9 . . . . 5 ((𝜑𝐷 ∈ ω) → 𝑧 ∈ V)
13 fvexg 5194 . . . . 5 (((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ V)
1410, 12, 13syl2anc 391 . . . 4 ((𝜑𝐷 ∈ ω) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ V)
1514alrimiv 1754 . . 3 ((𝜑𝐷 ∈ ω) → ∀𝑧((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ V)
16 frec2uz.1 . . . . . 6 (𝜑𝐶 ∈ ℤ)
17 uzid 8487 . . . . . 6 (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ𝐶))
1816, 17syl 14 . . . . 5 (𝜑𝐶 ∈ (ℤ𝐶))
19 uzrdg.a . . . . 5 (𝜑𝐴𝑆)
20 opelxp 4374 . . . . 5 (⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆))
2118, 19, 20sylanbrc 394 . . . 4 (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
2221adantr 261 . . 3 ((𝜑𝐷 ∈ ω) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
23 1st2nd2 5801 . . . . . . 7 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
24 fveq2 5178 . . . . . . . 8 (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩))
25 df-ov 5515 . . . . . . . 8 ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩)
2624, 25syl6eqr 2090 . . . . . . 7 (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)))
2723, 26syl 14 . . . . . 6 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)))
2827adantl 262 . . . . 5 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)))
29 xp1st 5792 . . . . . . 7 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (1st𝑧) ∈ (ℤ𝐶))
3029adantl 262 . . . . . 6 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (1st𝑧) ∈ (ℤ𝐶))
31 xp2nd 5793 . . . . . . 7 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (2nd𝑧) ∈ 𝑆)
3231adantl 262 . . . . . 6 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑆)
33 peano2uz 8526 . . . . . . . 8 ((1st𝑧) ∈ (ℤ𝐶) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
3430, 33syl 14 . . . . . . 7 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
35 uzrdg.f . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
3635ralrimivva 2401 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
3736ad2antrr 457 . . . . . . . 8 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
38 oveq1 5519 . . . . . . . . . . 11 (𝑥 = (1st𝑧) → (𝑥𝐹𝑦) = ((1st𝑧)𝐹𝑦))
3938eleq1d 2106 . . . . . . . . . 10 (𝑥 = (1st𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹𝑦) ∈ 𝑆))
40 oveq2 5520 . . . . . . . . . . 11 (𝑦 = (2nd𝑧) → ((1st𝑧)𝐹𝑦) = ((1st𝑧)𝐹(2nd𝑧)))
4140eleq1d 2106 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → (((1st𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
4239, 41rspc2v 2662 . . . . . . . . 9 (((1st𝑧) ∈ (ℤ𝐶) ∧ (2nd𝑧) ∈ 𝑆) → (∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
4330, 32, 42syl2anc 391 . . . . . . . 8 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
4437, 43mpd 13 . . . . . . 7 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆)
45 opelxp 4374 . . . . . . 7 (⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((1st𝑧) + 1) ∈ (ℤ𝐶) ∧ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
4634, 44, 45sylanbrc 394 . . . . . 6 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆))
47 oveq1 5519 . . . . . . . 8 (𝑥 = (1st𝑧) → (𝑥 + 1) = ((1st𝑧) + 1))
4847, 38opeq12d 3557 . . . . . . 7 (𝑥 = (1st𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩)
4940opeq2d 3556 . . . . . . 7 (𝑦 = (2nd𝑧) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
50 eqid 2040 . . . . . . 7 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
5148, 49, 50ovmpt2g 5635 . . . . . 6 (((1st𝑧) ∈ (ℤ𝐶) ∧ (2nd𝑧) ∈ 𝑆 ∧ ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
5230, 32, 46, 51syl3anc 1135 . . . . 5 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
5328, 52eqtrd 2072 . . . 4 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
5453, 46eqeltrd 2114 . . 3 (((𝜑𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
55 simpr 103 . . 3 ((𝜑𝐷 ∈ ω) → 𝐷 ∈ ω)
5615, 22, 54, 55freccl 5993 . 2 ((𝜑𝐷 ∈ ω) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝐷) ∈ ((ℤ𝐶) × 𝑆))
572, 56syl5eqel 2124 1 ((𝜑𝐷 ∈ ω) → (𝑅𝐷) ∈ ((ℤ𝐶) × 𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  Vcvv 2557  ⟨cop 3378   ↦ cmpt 3818  ωcom 4313   × cxp 4343  ‘cfv 4902  (class class class)co 5512   ↦ cmpt2 5514  1st c1st 5765  2nd c2nd 5766  freccfrec 5977  1c1 6890   + caddc 6892  ℤcz 8245  ℤ≥cuz 8473 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474 This theorem is referenced by:  frec2uzrdg  9195  frecuzrdgfn  9198  frecuzrdgcl  9199  frecuzrdgsuc  9201
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