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Theorem frec2uzrdg 9195
 Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ0) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. This lemma shows that evaluating 𝑅 at an element of ω gives an ordered pair whose first element is the index (translated from ω to (ℤ≥‘𝐶)). See comment in frec2uz0d 9185 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-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⟩)
uzrdg.b (𝜑𝐵 ∈ ω)
Assertion
Ref Expression
frec2uzrdg (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐶,𝑦   𝑦,𝐺   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem frec2uzrdg
Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzrdg.b . 2 (𝜑𝐵 ∈ ω)
2 fveq2 5178 . . . . 5 (𝑧 = 𝐵 → (𝑅𝑧) = (𝑅𝐵))
3 fveq2 5178 . . . . . 6 (𝑧 = 𝐵 → (𝐺𝑧) = (𝐺𝐵))
42fveq2d 5182 . . . . . 6 (𝑧 = 𝐵 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝐵)))
53, 4opeq12d 3557 . . . . 5 (𝑧 = 𝐵 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
62, 5eqeq12d 2054 . . . 4 (𝑧 = 𝐵 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
76imbi2d 219 . . 3 (𝑧 = 𝐵 → ((𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩) ↔ (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)))
8 fveq2 5178 . . . . 5 (𝑧 = ∅ → (𝑅𝑧) = (𝑅‘∅))
9 fveq2 5178 . . . . . 6 (𝑧 = ∅ → (𝐺𝑧) = (𝐺‘∅))
108fveq2d 5182 . . . . . 6 (𝑧 = ∅ → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘∅)))
119, 10opeq12d 3557 . . . . 5 (𝑧 = ∅ → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
128, 11eqeq12d 2054 . . . 4 (𝑧 = ∅ → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
13 fveq2 5178 . . . . 5 (𝑧 = 𝑣 → (𝑅𝑧) = (𝑅𝑣))
14 fveq2 5178 . . . . . 6 (𝑧 = 𝑣 → (𝐺𝑧) = (𝐺𝑣))
1513fveq2d 5182 . . . . . 6 (𝑧 = 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝑣)))
1614, 15opeq12d 3557 . . . . 5 (𝑧 = 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
1713, 16eqeq12d 2054 . . . 4 (𝑧 = 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
18 fveq2 5178 . . . . 5 (𝑧 = suc 𝑣 → (𝑅𝑧) = (𝑅‘suc 𝑣))
19 fveq2 5178 . . . . . 6 (𝑧 = suc 𝑣 → (𝐺𝑧) = (𝐺‘suc 𝑣))
2018fveq2d 5182 . . . . . 6 (𝑧 = suc 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
2119, 20opeq12d 3557 . . . . 5 (𝑧 = suc 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
2218, 21eqeq12d 2054 . . . 4 (𝑧 = suc 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
23 uzrdg.2 . . . . . . 7 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
2423fveq1i 5179 . . . . . 6 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
25 frec2uz.1 . . . . . . . 8 (𝜑𝐶 ∈ ℤ)
26 uzrdg.a . . . . . . . 8 (𝜑𝐴𝑆)
27 opexg 3964 . . . . . . . 8 ((𝐶 ∈ ℤ ∧ 𝐴𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
2825, 26, 27syl2anc 391 . . . . . . 7 (𝜑 → ⟨𝐶, 𝐴⟩ ∈ V)
29 frec0g 5983 . . . . . . 7 (⟨𝐶, 𝐴⟩ ∈ V → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
3028, 29syl 14 . . . . . 6 (𝜑 → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
3124, 30syl5eq 2084 . . . . 5 (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
32 frec2uz.2 . . . . . . 7 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
3325, 32frec2uz0d 9185 . . . . . 6 (𝜑 → (𝐺‘∅) = 𝐶)
3431fveq2d 5182 . . . . . . 7 (𝜑 → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩))
35 uzid 8487 . . . . . . . . 9 (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ𝐶))
3625, 35syl 14 . . . . . . . 8 (𝜑𝐶 ∈ (ℤ𝐶))
37 op2ndg 5778 . . . . . . . 8 ((𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆) → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3836, 26, 37syl2anc 391 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3934, 38eqtrd 2072 . . . . . 6 (𝜑 → (2nd ‘(𝑅‘∅)) = 𝐴)
4033, 39opeq12d 3557 . . . . 5 (𝜑 → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴⟩)
4131, 40eqtr4d 2075 . . . 4 (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
42 zex 8254 . . . . . . . . . . . . . . . 16 ℤ ∈ V
43 uzssz 8492 . . . . . . . . . . . . . . . 16 (ℤ𝐶) ⊆ ℤ
4442, 43ssexi 3895 . . . . . . . . . . . . . . 15 (ℤ𝐶) ∈ V
4544a1i 9 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ ω) → (ℤ𝐶) ∈ V)
46 uzrdg.s . . . . . . . . . . . . . . 15 (𝜑𝑆𝑉)
4746adantr 261 . . . . . . . . . . . . . 14 ((𝜑𝑣 ∈ ω) → 𝑆𝑉)
48 mpt2exga 5835 . . . . . . . . . . . . . 14 (((ℤ𝐶) ∈ V ∧ 𝑆𝑉) → (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) ∈ V)
4945, 47, 48syl2anc 391 . . . . . . . . . . . . 13 ((𝜑𝑣 ∈ ω) → (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) ∈ V)
50 vex 2560 . . . . . . . . . . . . . 14 𝑧 ∈ V
5150a1i 9 . . . . . . . . . . . . 13 ((𝜑𝑣 ∈ ω) → 𝑧 ∈ V)
52 fvexg 5194 . . . . . . . . . . . . 13 (((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ V)
5349, 51, 52syl2anc 391 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ V)
5453alrimiv 1754 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ∀𝑧((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ V)
5528adantr 261 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ⟨𝐶, 𝐴⟩ ∈ V)
56 simpr 103 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → 𝑣 ∈ ω)
57 frecsuc 5991 . . . . . . . . . . 11 ((∀𝑧((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ V ∧ ⟨𝐶, 𝐴⟩ ∈ V ∧ 𝑣 ∈ ω) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)))
5854, 55, 56, 57syl3anc 1135 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)))
5923fveq1i 5179 . . . . . . . . . 10 (𝑅‘suc 𝑣) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣)
6023fveq1i 5179 . . . . . . . . . . 11 (𝑅𝑣) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)
6160fveq2i 5181 . . . . . . . . . 10 ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣))
6258, 59, 613eqtr4g 2097 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
6362adantr 261 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
64 fveq2 5178 . . . . . . . . 9 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
65 df-ov 5515 . . . . . . . . . 10 ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
6625adantr 261 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐶 ∈ ℤ)
6766, 32, 56frec2uzuzd 9188 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → (𝐺𝑣) ∈ (ℤ𝐶))
68 uzrdg.f . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
6925, 32, 46, 26, 68, 23frecuzrdgrrn 9194 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → (𝑅𝑣) ∈ ((ℤ𝐶) × 𝑆))
70 xp2nd 5793 . . . . . . . . . . . 12 ((𝑅𝑣) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅𝑣)) ∈ 𝑆)
7169, 70syl 14 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → (2nd ‘(𝑅𝑣)) ∈ 𝑆)
72 peano2uz 8526 . . . . . . . . . . . . 13 ((𝐺𝑣) ∈ (ℤ𝐶) → ((𝐺𝑣) + 1) ∈ (ℤ𝐶))
7367, 72syl 14 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣) + 1) ∈ (ℤ𝐶))
7468caovclg 5653 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
7574adantlr 446 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
7675, 67, 71caovcld 5654 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆)
77 opelxp 4374 . . . . . . . . . . . 12 (⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((𝐺𝑣) + 1) ∈ (ℤ𝐶) ∧ ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆))
7873, 76, 77sylanbrc 394 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆))
79 oveq1 5519 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → (𝑤 + 1) = ((𝐺𝑣) + 1))
80 oveq1 5519 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → (𝑤𝐹𝑧) = ((𝐺𝑣)𝐹𝑧))
8179, 80opeq12d 3557 . . . . . . . . . . . 12 (𝑤 = (𝐺𝑣) → ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩)
82 oveq2 5520 . . . . . . . . . . . . 13 (𝑧 = (2nd ‘(𝑅𝑣)) → ((𝐺𝑣)𝐹𝑧) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
8382opeq2d 3556 . . . . . . . . . . . 12 (𝑧 = (2nd ‘(𝑅𝑣)) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
84 oveq1 5519 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥 + 1) = (𝑤 + 1))
85 oveq1 5519 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
8684, 85opeq12d 3557 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩)
87 oveq2 5520 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
8887opeq2d 3556 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
8986, 88cbvmpt2v 5584 . . . . . . . . . . . 12 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ (ℤ𝐶), 𝑧𝑆 ↦ ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
9081, 83, 89ovmpt2g 5635 . . . . . . . . . . 11 (((𝐺𝑣) ∈ (ℤ𝐶) ∧ (2nd ‘(𝑅𝑣)) ∈ 𝑆 ∧ ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆)) → ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
9167, 71, 78, 90syl3anc 1135 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
9265, 91syl5eqr 2086 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
9364, 92sylan9eqr 2094 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
9463, 93eqtrd 2072 . . . . . . 7 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
9566, 32, 56frec2uzsucd 9187 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
9695adantr 261 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
9794fveq2d 5182 . . . . . . . . 9 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩))
9866, 32, 56frec2uzzd 9186 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → (𝐺𝑣) ∈ ℤ)
9998peano2zd 8363 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣) + 1) ∈ ℤ)
10099adantr 261 . . . . . . . . . 10 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝐺𝑣) + 1) ∈ ℤ)
10176adantr 261 . . . . . . . . . 10 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆)
102 op2ndg 5778 . . . . . . . . . 10 ((((𝐺𝑣) + 1) ∈ ℤ ∧ ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆) → (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
103100, 101, 102syl2anc 391 . . . . . . . . 9 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
10497, 103eqtrd 2072 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
10596, 104opeq12d 3557 . . . . . . 7 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
10694, 105eqtr4d 2075 . . . . . 6 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
107106ex 108 . . . . 5 ((𝜑𝑣 ∈ ω) → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
108107expcom 109 . . . 4 (𝑣 ∈ ω → (𝜑 → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
10912, 17, 22, 41, 108finds2 4324 . . 3 (𝑧 ∈ ω → (𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩))
1107, 109vtoclga 2619 . 2 (𝐵 ∈ ω → (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
1111, 110mpcom 32 1 (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ∅c0 3224  ⟨cop 3378   ↦ cmpt 3818  suc csuc 4102  ωcom 4313   × cxp 4343  ‘cfv 4902  (class class class)co 5512   ↦ cmpt2 5514  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:  frecuzrdglem  9197  frecuzrdgfn  9198  frecuzrdgsuc  9201
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