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Theorem frecuzrdgg 9550
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 5229 . . . . . 6 (𝑤 = ∅ → (𝑅𝑤) = (𝑅‘∅))
32fveq2d 5233 . . . . 5 (𝑤 = ∅ → (1st ‘(𝑅𝑤)) = (1st ‘(𝑅‘∅)))
4 fveq2 5229 . . . . 5 (𝑤 = ∅ → (𝐺𝑤) = (𝐺‘∅))
53, 4eqeq12d 2097 . . . 4 (𝑤 = ∅ → ((1st ‘(𝑅𝑤)) = (𝐺𝑤) ↔ (1st ‘(𝑅‘∅)) = (𝐺‘∅)))
65imbi2d 228 . . 3 (𝑤 = ∅ → ((𝜑 → (1st ‘(𝑅𝑤)) = (𝐺𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))))
7 fveq2 5229 . . . . . 6 (𝑤 = 𝑘 → (𝑅𝑤) = (𝑅𝑘))
87fveq2d 5233 . . . . 5 (𝑤 = 𝑘 → (1st ‘(𝑅𝑤)) = (1st ‘(𝑅𝑘)))
9 fveq2 5229 . . . . 5 (𝑤 = 𝑘 → (𝐺𝑤) = (𝐺𝑘))
108, 9eqeq12d 2097 . . . 4 (𝑤 = 𝑘 → ((1st ‘(𝑅𝑤)) = (𝐺𝑤) ↔ (1st ‘(𝑅𝑘)) = (𝐺𝑘)))
1110imbi2d 228 . . 3 (𝑤 = 𝑘 → ((𝜑 → (1st ‘(𝑅𝑤)) = (𝐺𝑤)) ↔ (𝜑 → (1st ‘(𝑅𝑘)) = (𝐺𝑘))))
12 fveq2 5229 . . . . . 6 (𝑤 = suc 𝑘 → (𝑅𝑤) = (𝑅‘suc 𝑘))
1312fveq2d 5233 . . . . 5 (𝑤 = suc 𝑘 → (1st ‘(𝑅𝑤)) = (1st ‘(𝑅‘suc 𝑘)))
14 fveq2 5229 . . . . 5 (𝑤 = suc 𝑘 → (𝐺𝑤) = (𝐺‘suc 𝑘))
1513, 14eqeq12d 2097 . . . 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 5229 . . . . . 6 (𝑤 = 𝑁 → (𝑅𝑤) = (𝑅𝑁))
1817fveq2d 5233 . . . . 5 (𝑤 = 𝑁 → (1st ‘(𝑅𝑤)) = (1st ‘(𝑅𝑁)))
19 fveq2 5229 . . . . 5 (𝑤 = 𝑁 → (𝐺𝑤) = (𝐺𝑁))
2018, 19eqeq12d 2097 . . . 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 5828 . . . . 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 5230 . . . . . 6 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
28 opexg 4011 . . . . . . . 8 ((𝐶 ∈ ℤ ∧ 𝐴𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
29 frec0g 6066 . . . . . . . 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 2127 . . . . 5 (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
3332fveq2d 5233 . . . 4 (𝜑 → (1st ‘(𝑅‘∅)) = (1st ‘⟨𝐶, 𝐴⟩))
34 frecuzrdgg.g . . . . 5 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
3522, 34frec2uz0d 9533 . . . 4 (𝜑 → (𝐺‘∅) = 𝐶)
3625, 33, 353eqtr4d 2125 . . 3 (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))
3722, 34frec2uzf1od 9540 . . . . . . . . . . 11 (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))
38 f1of 5177 . . . . . . . . . . 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 5355 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝐺𝑘) ∈ (ℤ𝐶))
43 peano2uz 8804 . . . . . . . 8 ((𝐺𝑘) ∈ (ℤ𝐶) → ((𝐺𝑘) + 1) ∈ (ℤ𝐶))
4442, 43syl 14 . . . . . . 7 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝐺𝑘) + 1) ∈ (ℤ𝐶))
45 oveq2 5571 . . . . . . . . 9 (𝑦 = (2nd ‘(𝑅𝑘)) → ((𝐺𝑘)𝐹𝑦) = ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘))))
4645eleq1d 2151 . . . . . . . 8 (𝑦 = (2nd ‘(𝑅𝑘)) → (((𝐺𝑘)𝐹𝑦) ∈ 𝑆 ↔ ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘))) ∈ 𝑆))
47 oveq1 5570 . . . . . . . . . . 11 (𝑥 = (𝐺𝑘) → (𝑥𝐹𝑦) = ((𝐺𝑘)𝐹𝑦))
4847eleq1d 2151 . . . . . . . . . 10 (𝑥 = (𝐺𝑘) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((𝐺𝑘)𝐹𝑦) ∈ 𝑆))
4948ralbidv 2373 . . . . . . . . 9 (𝑥 = (𝐺𝑘) → (∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑆 ((𝐺𝑘)𝐹𝑦) ∈ 𝑆))
50 frecuzrdgrclt.f . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
5150ralrimivva 2448 . . . . . . . . . 10 (𝜑 → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5251ad2antlr 473 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5349, 52, 42rspcdva 2715 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ∀𝑦𝑆 ((𝐺𝑘)𝐹𝑦) ∈ 𝑆)
54 frecuzrdgrclt.t . . . . . . . . . . . 12 (𝜑𝑆𝑇)
5522, 23, 54, 50, 26frecuzrdgrclt 9549 . . . . . . . . . . 11 (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
5655ad2antlr 473 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → 𝑅:ω⟶((ℤ𝐶) × 𝑆))
5756, 41ffvelrnd 5355 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅𝑘) ∈ ((ℤ𝐶) × 𝑆))
58 xp2nd 5844 . . . . . . . . 9 ((𝑅𝑘) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅𝑘)) ∈ 𝑆)
5957, 58syl 14 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (2nd ‘(𝑅𝑘)) ∈ 𝑆)
6046, 53, 59rspcdva 2715 . . . . . . 7 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘))) ∈ 𝑆)
61 op1stg 5828 . . . . . . 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 5852 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6463adantl 271 . . . . . . . . . . . . . . 15 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6564fveq2d 5233 . . . . . . . . . . . . . 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 5566 . . . . . . . . . . . . . . . 16 ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩)
67 xp1st 5843 . . . . . . . . . . . . . . . . . 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 5844 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (2nd𝑧) ∈ 𝑆)
7170adantl 271 . . . . . . . . . . . . . . . . . 18 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑆)
7269, 71sseldd 3009 . . . . . . . . . . . . . . . . 17 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑇)
73 peano2uz 8804 . . . . . . . . . . . . . . . . . . 19 ((1st𝑧) ∈ (ℤ𝐶) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
7468, 73syl 14 . . . . . . . . . . . . . . . . . 18 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
75 oveq2 5571 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (2nd𝑧) → ((1st𝑧)𝐹𝑦) = ((1st𝑧)𝐹(2nd𝑧)))
7675eleq1d 2151 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (2nd𝑧) → (((1st𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
77 oveq1 5570 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (1st𝑧) → (𝑥𝐹𝑦) = ((1st𝑧)𝐹𝑦))
7877eleq1d 2151 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (1st𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹𝑦) ∈ 𝑆))
7978ralbidv 2373 . . . . . . . . . . . . . . . . . . . 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 2715 . . . . . . . . . . . . . . . . . . 19 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑦𝑆 ((1st𝑧)𝐹𝑦) ∈ 𝑆)
8276, 81, 71rspcdva 2715 . . . . . . . . . . . . . . . . . 18 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆)
83 opelxpi 4422 . . . . . . . . . . . . . . . . . 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 5570 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (1st𝑧) → (𝑥 + 1) = ((1st𝑧) + 1))
8685, 77opeq12d 3598 . . . . . . . . . . . . . . . . . 18 (𝑥 = (1st𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩)
8775opeq2d 3597 . . . . . . . . . . . . . . . . . 18 (𝑦 = (2nd𝑧) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
88 eqid 2083 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
8986, 87, 88ovmpt2g 5686 . . . . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . . . . 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 2129 . . . . . . . . . . . . . . 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 2159 . . . . . . . . . . . . . 14 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) ∈ ((ℤ𝐶) × 𝑆))
9365, 92eqeltrd 2159 . . . . . . . . . . . . 13 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
9493ralrimiva 2439 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
95 uzid 8766 . . . . . . . . . . . . . . 15 (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ𝐶))
9622, 95syl 14 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ (ℤ𝐶))
97 opelxpi 4422 . . . . . . . . . . . . . 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 6076 . . . . . . . . . . . 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 1170 . . . . . . . . . . 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 5230 . . . . . . . . . . 11 (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘)
10326fveq1i 5230 . . . . . . . . . . . 12 (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)
104103fveq2i 5232 . . . . . . . . . . 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 2140 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑘)))
106 1st2nd2 5852 . . . . . . . . . . . 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 5233 . . . . . . . . . 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 2115 . . . . . . . . 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 3596 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩ = ⟨(𝐺𝑘), (2nd ‘(𝑅𝑘))⟩)
112111fveq2d 5233 . . . . . . . . 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 2115 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑘), (2nd ‘(𝑅𝑘))⟩))
114 df-ov 5566 . . . . . . . . 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 3009 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (2nd ‘(𝑅𝑘)) ∈ 𝑇)
117 opelxpi 4422 . . . . . . . . . . 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 5570 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑘) → (𝑥 + 1) = ((𝐺𝑘) + 1))
120119, 47opeq12d 3598 . . . . . . . . . . 11 (𝑥 = (𝐺𝑘) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹𝑦)⟩)
12145opeq2d 3597 . . . . . . . . . . 11 (𝑦 = (2nd ‘(𝑅𝑘)) → ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹𝑦)⟩ = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
122120, 121, 88ovmpt2g 5686 . . . . . . . . . 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 1170 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝐺𝑘)(𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑘))) = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
124114, 123syl5eqr 2129 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑘), (2nd ‘(𝑅𝑘))⟩) = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
125113, 124eqtrd 2115 . . . . . . 7 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝑅‘suc 𝑘) = ⟨((𝐺𝑘) + 1), ((𝐺𝑘)𝐹(2nd ‘(𝑅𝑘)))⟩)
126125fveq2d 5233 . . . . . 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 9535 . . . . . 6 (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅𝑘)) = (𝐺𝑘)) → (𝐺‘suc 𝑘) = ((𝐺𝑘) + 1))
12962, 126, 1283eqtr4d 2125 . . . . 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 4369 . 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 1285  wcel 1434  wral 2353  Vcvv 2610  wss 2982  c0 3267  cop 3419  cmpt 3859  suc csuc 4148  ωcom 4359   × cxp 4389  wf 4948  1-1-ontowf1o 4951  cfv 4952  (class class class)co 5563  cmpt2 5565  1st c1st 5816  2nd c2nd 5817  freccfrec 6059  1c1 7096   + caddc 7098  cz 8484  cuz 8752
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753
This theorem is referenced by:  frecuzrdgdomlem  9551  frecuzrdgfunlem  9553  frecuzrdgsuctlem  9557
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