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Theorem frecuzrdgg 9788
Description: Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating 𝑅 at a natural number gives an ordered pair whose first element is the mapping of that natural number via 𝐺. (Contributed by Jim Kingdon, 23-Apr-2022.)
Hypotheses
Ref Expression
frecuzrdgrclt.c (𝜑𝐶 ∈ ℤ)
frecuzrdgrclt.a (𝜑𝐴𝑆)
frecuzrdgrclt.t (𝜑𝑆𝑇)
frecuzrdgrclt.f ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrclt.r 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgg.n (𝜑𝑁 ∈ ω)
frecuzrdgg.g 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
Assertion
Ref Expression
frecuzrdgg (𝜑 → (1st ‘(𝑅𝑁)) = (𝐺𝑁))
Distinct variable groups:   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem frecuzrdgg
Dummy variables 𝑧 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecuzrdgg.n . 2 (𝜑𝑁 ∈ ω)
2 fveq2 5289 . . . . . 6 (𝑤 = ∅ → (𝑅𝑤) = (𝑅‘∅))
32fveq2d 5293 . . . . 5 (𝑤 = ∅ → (1st ‘(𝑅𝑤)) = (1st ‘(𝑅‘∅)))
4 fveq2 5289 . . . . 5 (𝑤 = ∅ → (𝐺𝑤) = (𝐺‘∅))
53, 4eqeq12d 2102 . . . 4 (𝑤 = ∅ → ((1st ‘(𝑅𝑤)) = (𝐺𝑤) ↔ (1st ‘(𝑅‘∅)) = (𝐺‘∅)))
65imbi2d 228 . . 3 (𝑤 = ∅ → ((𝜑 → (1st ‘(𝑅𝑤)) = (𝐺𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))))
7 fveq2 5289 . . . . . 6 (𝑤 = 𝑘 → (𝑅𝑤) = (𝑅𝑘))
87fveq2d 5293 . . . . 5 (𝑤 = 𝑘 → (1st ‘(𝑅𝑤)) = (1st ‘(𝑅𝑘)))
9 fveq2 5289 . . . . 5 (𝑤 = 𝑘 → (𝐺𝑤) = (𝐺𝑘))
108, 9eqeq12d 2102 . . . 4 (𝑤 = 𝑘 → ((1st ‘(𝑅𝑤)) = (𝐺𝑤) ↔ (1st ‘(𝑅𝑘)) = (𝐺𝑘)))
1110imbi2d 228 . . 3 (𝑤 = 𝑘 → ((𝜑 → (1st ‘(𝑅𝑤)) = (𝐺𝑤)) ↔ (𝜑 → (1st ‘(𝑅𝑘)) = (𝐺𝑘))))
12 fveq2 5289 . . . . . 6 (𝑤 = suc 𝑘 → (𝑅𝑤) = (𝑅‘suc 𝑘))
1312fveq2d 5293 . . . . 5 (𝑤 = suc 𝑘 → (1st ‘(𝑅𝑤)) = (1st ‘(𝑅‘suc 𝑘)))
14 fveq2 5289 . . . . 5 (𝑤 = suc 𝑘 → (𝐺𝑤) = (𝐺‘suc 𝑘))
1513, 14eqeq12d 2102 . . . 4 (𝑤 = suc 𝑘 → ((1st ‘(𝑅𝑤)) = (𝐺𝑤) ↔ (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘)))
1615imbi2d 228 . . 3 (𝑤 = suc 𝑘 → ((𝜑 → (1st ‘(𝑅𝑤)) = (𝐺𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))))
17 fveq2 5289 . . . . . 6 (𝑤 = 𝑁 → (𝑅𝑤) = (𝑅𝑁))
1817fveq2d 5293 . . . . 5 (𝑤 = 𝑁 → (1st ‘(𝑅𝑤)) = (1st ‘(𝑅𝑁)))
19 fveq2 5289 . . . . 5 (𝑤 = 𝑁 → (𝐺𝑤) = (𝐺𝑁))
2018, 19eqeq12d 2102 . . . 4 (𝑤 = 𝑁 → ((1st ‘(𝑅𝑤)) = (𝐺𝑤) ↔ (1st ‘(𝑅𝑁)) = (𝐺𝑁)))
2120imbi2d 228 . . 3 (𝑤 = 𝑁 → ((𝜑 → (1st ‘(𝑅𝑤)) = (𝐺𝑤)) ↔ (𝜑 → (1st ‘(𝑅𝑁)) = (𝐺𝑁))))
22 frecuzrdgrclt.c . . . . 5 (𝜑𝐶 ∈ ℤ)
23 frecuzrdgrclt.a . . . . 5 (𝜑𝐴𝑆)
24 op1stg 5903 . . . . 5 ((𝐶 ∈ ℤ ∧ 𝐴𝑆) → (1st ‘⟨𝐶, 𝐴⟩) = 𝐶)
2522, 23, 24syl2anc 403 . . . 4 (𝜑 → (1st ‘⟨𝐶, 𝐴⟩) = 𝐶)
26 frecuzrdgrclt.r . . . . . . 7 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
2726fveq1i 5290 . . . . . 6 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
28 opexg 4046 . . . . . . . 8 ((𝐶 ∈ ℤ ∧ 𝐴𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
29 frec0g 6144 . . . . . . . 8 (⟨𝐶, 𝐴⟩ ∈ V → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
3028, 29syl 14 . . . . . . 7 ((𝐶 ∈ ℤ ∧ 𝐴𝑆) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
3122, 23, 30syl2anc 403 . . . . . 6 (𝜑 → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
3227, 31syl5eq 2132 . . . . 5 (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
3332fveq2d 5293 . . . 4 (𝜑 → (1st ‘(𝑅‘∅)) = (1st ‘⟨𝐶, 𝐴⟩))
34 frecuzrdgg.g . . . . 5 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
3522, 34frec2uz0d 9771 . . . 4 (𝜑 → (𝐺‘∅) = 𝐶)
3625, 33, 353eqtr4d 2130 . . 3 (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))
3722, 34frec2uzf1od 9778 . . . . . . . . . . 11 (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))
38 f1of 5237 . . . . . . . . . . 11 (𝐺:ω–1-1-onto→(ℤ𝐶) → 𝐺:ω⟶(ℤ𝐶))
3937, 38syl 14 . . . . . . . . . 10 (𝜑𝐺:ω⟶(ℤ𝐶))
4039ad2antlr 473 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → 𝐺:ω⟶(ℤ𝐶))
41 simpll 496 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → 𝑘 ∈ ω)
4240, 41ffvelrnd 5419 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝐺𝑘) ∈ (ℤ𝐶))
43 peano2uz 9040 . . . . . . . 8 ((𝐺𝑘) ∈ (ℤ𝐶) → ((𝐺𝑘) + 1) ∈ (ℤ𝐶))
4442, 43syl 14 . . . . . . 7 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝐺𝑘) + 1) ∈ (ℤ𝐶))
45 oveq2 5642 . . . . . . . . 9 (𝑦 = (2nd ‘(𝑅𝑘)) → ((𝐺𝑘)𝐹𝑦) = ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘))))
4645eleq1d 2156 . . . . . . . 8 (𝑦 = (2nd ‘(𝑅𝑘)) → (((𝐺𝑘)𝐹𝑦) ∈ 𝑆 ↔ ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘))) ∈ 𝑆))
47 oveq1 5641 . . . . . . . . . . 11 (𝑥 = (𝐺𝑘) → (𝑥𝐹𝑦) = ((𝐺𝑘)𝐹𝑦))
4847eleq1d 2156 . . . . . . . . . 10 (𝑥 = (𝐺𝑘) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((𝐺𝑘)𝐹𝑦) ∈ 𝑆))
4948ralbidv 2380 . . . . . . . . 9 (𝑥 = (𝐺𝑘) → (∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑆 ((𝐺𝑘)𝐹𝑦) ∈ 𝑆))
50 frecuzrdgrclt.f . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
5150ralrimivva 2455 . . . . . . . . . 10 (𝜑 → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5251ad2antlr 473 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5349, 52, 42rspcdva 2727 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ∀𝑦𝑆 ((𝐺𝑘)𝐹𝑦) ∈ 𝑆)
54 frecuzrdgrclt.t . . . . . . . . . . . 12 (𝜑𝑆𝑇)
5522, 23, 54, 50, 26frecuzrdgrclt 9787 . . . . . . . . . . 11 (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
5655ad2antlr 473 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → 𝑅:ω⟶((ℤ𝐶) × 𝑆))
5756, 41ffvelrnd 5419 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅𝑘) ∈ ((ℤ𝐶) × 𝑆))
58 xp2nd 5919 . . . . . . . . 9 ((𝑅𝑘) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅𝑘)) ∈ 𝑆)
5957, 58syl 14 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (2nd ‘(𝑅𝑘)) ∈ 𝑆)
6046, 53, 59rspcdva 2727 . . . . . . 7 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘))) ∈ 𝑆)
61 op1stg 5903 . . . . . . 7 ((((𝐺𝑘) + 1) ∈ (ℤ𝐶) ∧ ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘))) ∈ 𝑆) → (1st ‘⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩) = ((𝐺𝑘) + 1))
6244, 60, 61syl2anc 403 . . . . . 6 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (1st ‘⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩) = ((𝐺𝑘) + 1))
63 1st2nd2 5927 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6463adantl 271 . . . . . . . . . . . . . . 15 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6564fveq2d 5293 . . . . . . . . . . . . . 14 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩))
66 df-ov 5637 . . . . . . . . . . . . . . . 16 ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩)
67 xp1st 5918 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (1st𝑧) ∈ (ℤ𝐶))
6867adantl 271 . . . . . . . . . . . . . . . . 17 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (1st𝑧) ∈ (ℤ𝐶))
6954ad3antlr 477 . . . . . . . . . . . . . . . . . 18 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → 𝑆𝑇)
70 xp2nd 5919 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (2nd𝑧) ∈ 𝑆)
7170adantl 271 . . . . . . . . . . . . . . . . . 18 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑆)
7269, 71sseldd 3024 . . . . . . . . . . . . . . . . 17 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑇)
73 peano2uz 9040 . . . . . . . . . . . . . . . . . . 19 ((1st𝑧) ∈ (ℤ𝐶) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
7468, 73syl 14 . . . . . . . . . . . . . . . . . 18 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
75 oveq2 5642 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (2nd𝑧) → ((1st𝑧)𝐹𝑦) = ((1st𝑧)𝐹(2nd𝑧)))
7675eleq1d 2156 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (2nd𝑧) → (((1st𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
77 oveq1 5641 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (1st𝑧) → (𝑥𝐹𝑦) = ((1st𝑧)𝐹𝑦))
7877eleq1d 2156 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (1st𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹𝑦) ∈ 𝑆))
7978ralbidv 2380 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (1st𝑧) → (∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑆 ((1st𝑧)𝐹𝑦) ∈ 𝑆))
8051ad3antlr 477 . . . . . . . . . . . . . . . . . . . 20 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
8179, 80, 68rspcdva 2727 . . . . . . . . . . . . . . . . . . 19 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑦𝑆 ((1st𝑧)𝐹𝑦) ∈ 𝑆)
8276, 81, 71rspcdva 2727 . . . . . . . . . . . . . . . . . 18 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆)
83 opelxpi 4459 . . . . . . . . . . . . . . . . . 18 ((((1st𝑧) + 1) ∈ (ℤ𝐶) ∧ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆))
8474, 82, 83syl2anc 403 . . . . . . . . . . . . . . . . 17 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆))
85 oveq1 5641 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (1st𝑧) → (𝑥 + 1) = ((1st𝑧) + 1))
8685, 77opeq12d 3625 . . . . . . . . . . . . . . . . . 18 (𝑥 = (1st𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩)
8775opeq2d 3624 . . . . . . . . . . . . . . . . . 18 (𝑦 = (2nd𝑧) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
88 eqid 2088 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
8986, 87, 88ovmpt2g 5761 . . . . . . . . . . . . . . . . 17 (((1st𝑧) ∈ (ℤ𝐶) ∧ (2nd𝑧) ∈ 𝑇 ∧ ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
9068, 72, 84, 89syl3anc 1174 . . . . . . . . . . . . . . . 16 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
9166, 90syl5eqr 2134 . . . . . . . . . . . . . . 15 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
9291, 84eqeltrd 2164 . . . . . . . . . . . . . 14 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) ∈ ((ℤ𝐶) × 𝑆))
9365, 92eqeltrd 2164 . . . . . . . . . . . . 13 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
9493ralrimiva 2446 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
95 uzid 9002 . . . . . . . . . . . . . . 15 (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ𝐶))
9622, 95syl 14 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ (ℤ𝐶))
97 opelxpi 4459 . . . . . . . . . . . . . 14 ((𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
9896, 23, 97syl2anc 403 . . . . . . . . . . . . 13 (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
9998ad2antlr 473 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
100 frecsuc 6154 . . . . . . . . . . . 12 ((∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆) ∧ ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)))
10194, 99, 41, 100syl3anc 1174 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)))
10226fveq1i 5290 . . . . . . . . . . 11 (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘)
10326fveq1i 5290 . . . . . . . . . . . 12 (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)
104103fveq2i 5292 . . . . . . . . . . 11 ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘))
105101, 102, 1043eqtr4g 2145 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑘)))
106 1st2nd2 5927 . . . . . . . . . . . 12 ((𝑅𝑘) ∈ ((ℤ𝐶) × 𝑆) → (𝑅𝑘) = ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
10757, 106syl 14 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅𝑘) = ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
108107fveq2d 5293 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
109105, 108eqtrd 2120 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
110 simpr 108 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (1st ‘(𝑅𝑘)) = (𝐺𝑘))
111110opeq1d 3623 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩ = ⟨(𝐺𝑘), (2nd ‘(𝑅𝑘))⟩)
112111fveq2d 5293 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑘), (2nd ‘(𝑅𝑘))⟩))
113109, 112eqtrd 2120 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑘), (2nd ‘(𝑅𝑘))⟩))
114 df-ov 5637 . . . . . . . . 9 ((𝐺𝑘)(𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑘))) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑘), (2nd ‘(𝑅𝑘))⟩)
11554ad2antlr 473 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → 𝑆𝑇)
116115, 59sseldd 3024 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (2nd ‘(𝑅𝑘)) ∈ 𝑇)
117 opelxpi 4459 . . . . . . . . . . 11 ((((𝐺𝑘) + 1) ∈ (ℤ𝐶) ∧ ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘))) ∈ 𝑆) → ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝐶) × 𝑆))
11844, 60, 117syl2anc 403 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝐶) × 𝑆))
119 oveq1 5641 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑘) → (𝑥 + 1) = ((𝐺𝑘) + 1))
120119, 47opeq12d 3625 . . . . . . . . . . 11 (𝑥 = (𝐺𝑘) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹𝑦)⟩)
12145opeq2d 3624 . . . . . . . . . . 11 (𝑦 = (2nd ‘(𝑅𝑘)) → ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹𝑦)⟩ = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
122120, 121, 88ovmpt2g 5761 . . . . . . . . . 10 (((𝐺𝑘) ∈ (ℤ𝐶) ∧ (2nd ‘(𝑅𝑘)) ∈ 𝑇 ∧ ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝐶) × 𝑆)) → ((𝐺𝑘)(𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑘))) = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
12342, 116, 118, 122syl3anc 1174 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝐺𝑘)(𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑘))) = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
124114, 123syl5eqr 2134 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑘), (2nd ‘(𝑅𝑘))⟩) = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
125113, 124eqtrd 2120 . . . . . . 7 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅‘suc 𝑘) = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
126125fveq2d 5293 . . . . . 6 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (1st ‘(𝑅‘suc 𝑘)) = (1st ‘⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩))
12722ad2antlr 473 . . . . . . 7 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → 𝐶 ∈ ℤ)
128127, 34, 41frec2uzsucd 9773 . . . . . 6 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝐺‘suc 𝑘) = ((𝐺𝑘) + 1))
12962, 126, 1283eqtr4d 2130 . . . . 5 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))
130129exp31 356 . . . 4 (𝑘 ∈ ω → (𝜑 → ((1st ‘(𝑅𝑘)) = (𝐺𝑘) → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))))
131130a2d 26 . . 3 (𝑘 ∈ ω → ((𝜑 → (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝜑 → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))))
1326, 11, 16, 21, 36, 131finds 4405 . 2 (𝑁 ∈ ω → (𝜑 → (1st ‘(𝑅𝑁)) = (𝐺𝑁)))
1331, 132mpcom 36 1 (𝜑 → (1st ‘(𝑅𝑁)) = (𝐺𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  wral 2359  Vcvv 2619  wss 2997  c0 3284  cop 3444  cmpt 3891  suc csuc 4183  ωcom 4395   × cxp 4426  wf 4998  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  cmpt2 5636  1st c1st 5891  2nd c2nd 5892  freccfrec 6137  1c1 7330   + caddc 7332  cz 8720  cuz 8988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989
This theorem is referenced by:  frecuzrdgdomlem  9789  frecuzrdgfunlem  9791  frecuzrdgsuctlem  9795
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