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Theorem om2noseqrdg 28325
Description: A helper lemma for the value of a recursive definition generator on a surreal sequence with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. (Contributed by Scott Fenton, 18-Apr-2025.)
Hypotheses
Ref Expression
om2noseq.1 (𝜑𝐶 No )
om2noseq.2 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseq.3 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
noseqrdg.1 (𝜑𝐴𝑉)
noseqrdg.2 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
Assertion
Ref Expression
om2noseqrdg ((𝜑𝐵 ∈ ω) → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem om2noseqrdg
Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . . 5 (𝑧 = ∅ → (𝑅𝑧) = (𝑅‘∅))
2 fveq2 6907 . . . . . 6 (𝑧 = ∅ → (𝐺𝑧) = (𝐺‘∅))
3 2fveq3 6912 . . . . . 6 (𝑧 = ∅ → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘∅)))
42, 3opeq12d 4886 . . . . 5 (𝑧 = ∅ → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
51, 4eqeq12d 2751 . . . 4 (𝑧 = ∅ → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
65imbi2d 340 . . 3 (𝑧 = ∅ → ((𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩) ↔ (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)))
7 fveq2 6907 . . . . 5 (𝑧 = 𝑣 → (𝑅𝑧) = (𝑅𝑣))
8 fveq2 6907 . . . . . 6 (𝑧 = 𝑣 → (𝐺𝑧) = (𝐺𝑣))
9 2fveq3 6912 . . . . . 6 (𝑧 = 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝑣)))
108, 9opeq12d 4886 . . . . 5 (𝑧 = 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
117, 10eqeq12d 2751 . . . 4 (𝑧 = 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
1211imbi2d 340 . . 3 (𝑧 = 𝑣 → ((𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩) ↔ (𝜑 → (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)))
13 fveq2 6907 . . . . 5 (𝑧 = suc 𝑣 → (𝑅𝑧) = (𝑅‘suc 𝑣))
14 fveq2 6907 . . . . . 6 (𝑧 = suc 𝑣 → (𝐺𝑧) = (𝐺‘suc 𝑣))
15 2fveq3 6912 . . . . . 6 (𝑧 = suc 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
1614, 15opeq12d 4886 . . . . 5 (𝑧 = suc 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
1713, 16eqeq12d 2751 . . . 4 (𝑧 = suc 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
1817imbi2d 340 . . 3 (𝑧 = suc 𝑣 → ((𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩) ↔ (𝜑 → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
19 fveq2 6907 . . . . 5 (𝑧 = 𝐵 → (𝑅𝑧) = (𝑅𝐵))
20 fveq2 6907 . . . . . 6 (𝑧 = 𝐵 → (𝐺𝑧) = (𝐺𝐵))
21 2fveq3 6912 . . . . . 6 (𝑧 = 𝐵 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝐵)))
2220, 21opeq12d 4886 . . . . 5 (𝑧 = 𝐵 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
2319, 22eqeq12d 2751 . . . 4 (𝑧 = 𝐵 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
2423imbi2d 340 . . 3 (𝑧 = 𝐵 → ((𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩) ↔ (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)))
25 noseqrdg.2 . . . . . 6 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
2625fveq1d 6909 . . . . 5 (𝜑 → (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅))
27 opex 5475 . . . . . 6 𝐶, 𝐴⟩ ∈ V
28 fr0g 8475 . . . . . 6 (⟨𝐶, 𝐴⟩ ∈ V → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩)
2927, 28ax-mp 5 . . . . 5 ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴
3026, 29eqtrdi 2791 . . . 4 (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
31 om2noseq.1 . . . . . 6 (𝜑𝐶 No )
32 om2noseq.2 . . . . . 6 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3331, 32om2noseq0 28317 . . . . 5 (𝜑 → (𝐺‘∅) = 𝐶)
3430fveq2d 6911 . . . . . 6 (𝜑 → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩))
35 noseqrdg.1 . . . . . . 7 (𝜑𝐴𝑉)
36 op2ndg 8026 . . . . . . 7 ((𝐶 No 𝐴𝑉) → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3731, 35, 36syl2anc 584 . . . . . 6 (𝜑 → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3834, 37eqtrd 2775 . . . . 5 (𝜑 → (2nd ‘(𝑅‘∅)) = 𝐴)
3933, 38opeq12d 4886 . . . 4 (𝜑 → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴⟩)
4030, 39eqtr4d 2778 . . 3 (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
41 frsuc 8476 . . . . . . . . . . 11 (𝑣 ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
4241adantl 481 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
4325fveq1d 6909 . . . . . . . . . . 11 (𝜑 → (𝑅‘suc 𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣))
4443adantr 480 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣))
4525fveq1d 6909 . . . . . . . . . . . 12 (𝜑 → (𝑅𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣))
4645fveq2d 6911 . . . . . . . . . . 11 (𝜑 → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
4746adantr 480 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
4842, 44, 473eqtr4d 2785 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
4948adantrr 717 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)) → (𝑅‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
50 fveq2 6907 . . . . . . . . . 10 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
51 df-ov 7434 . . . . . . . . . . 11 ((𝐺𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
52 fvex 6920 . . . . . . . . . . . 12 (𝐺𝑣) ∈ V
53 fvex 6920 . . . . . . . . . . . 12 (2nd ‘(𝑅𝑣)) ∈ V
54 oveq1 7438 . . . . . . . . . . . . . 14 (𝑤 = (𝐺𝑣) → (𝑤 +s 1s ) = ((𝐺𝑣) +s 1s ))
55 oveq1 7438 . . . . . . . . . . . . . 14 (𝑤 = (𝐺𝑣) → (𝑤𝐹𝑧) = ((𝐺𝑣)𝐹𝑧))
5654, 55opeq12d 4886 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → ⟨(𝑤 +s 1s ), (𝑤𝐹𝑧)⟩ = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹𝑧)⟩)
57 oveq2 7439 . . . . . . . . . . . . . 14 (𝑧 = (2nd ‘(𝑅𝑣)) → ((𝐺𝑣)𝐹𝑧) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
5857opeq2d 4885 . . . . . . . . . . . . 13 (𝑧 = (2nd ‘(𝑅𝑣)) → ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹𝑧)⟩ = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
59 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑥 +s 1s ) = (𝑤 +s 1s ))
60 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
6159, 60opeq12d 4886 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 +s 1s ), (𝑤𝐹𝑦)⟩)
62 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
6362opeq2d 4885 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ⟨(𝑤 +s 1s ), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 +s 1s ), (𝑤𝐹𝑧)⟩)
6461, 63cbvmpov 7528 . . . . . . . . . . . . 13 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑤 +s 1s ), (𝑤𝐹𝑧)⟩)
65 opex 5475 . . . . . . . . . . . . 13 ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ V
6656, 58, 64, 65ovmpo 7593 . . . . . . . . . . . 12 (((𝐺𝑣) ∈ V ∧ (2nd ‘(𝑅𝑣)) ∈ V) → ((𝐺𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
6752, 53, 66mp2an 692 . . . . . . . . . . 11 ((𝐺𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩
6851, 67eqtr3i 2765 . . . . . . . . . 10 ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩
6950, 68eqtrdi 2791 . . . . . . . . 9 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
7069ad2antll 729 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
7149, 70eqtrd 2775 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)) → (𝑅‘suc 𝑣) = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
7231adantr 480 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → 𝐶 No )
7332adantr 480 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
74 simpr 484 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → 𝑣 ∈ ω)
7572, 73, 74om2noseqsuc 28318 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝐺‘suc 𝑣) = ((𝐺𝑣) +s 1s ))
7675adantrr 717 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)) → (𝐺‘suc 𝑣) = ((𝐺𝑣) +s 1s ))
7771fveq2d 6911 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩))
78 ovex 7464 . . . . . . . . . 10 ((𝐺𝑣) +s 1s ) ∈ V
79 ovex 7464 . . . . . . . . . 10 ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ V
8078, 79op2nd 8022 . . . . . . . . 9 (2nd ‘⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))
8177, 80eqtrdi 2791 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
8276, 81opeq12d 4886 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺𝑣) +s 1s ), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
8371, 82eqtr4d 2778 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
8483exp32 420 . . . . 5 (𝜑 → (𝑣 ∈ ω → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
8584com12 32 . . . 4 (𝑣 ∈ ω → (𝜑 → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
8685a2d 29 . . 3 (𝑣 ∈ ω → ((𝜑 → (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝜑 → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
876, 12, 18, 24, 40, 86finds 7919 . 2 (𝐵 ∈ ω → (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
8887impcom 407 1 ((𝜑𝐵 ∈ ω) → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cop 4637  cmpt 5231  cres 5691  cima 5692  suc csuc 6388  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  2nd c2nd 8012  reccrdg 8448   No csur 27699   1s c1s 27883   +s cadds 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449
This theorem is referenced by:  noseqrdglem  28326  noseqrdgfn  28327  noseqrdgsuc  28329
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