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Theorem finxpreclem4 33361
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem4.1 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
Assertion
Ref Expression
finxpreclem4 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁))
Distinct variable groups:   𝑛,𝑁,𝑥   𝑈,𝑛,𝑥   𝑦,𝑛,𝑥
Allowed substitution hints:   𝑈(𝑦)   𝐹(𝑥,𝑦,𝑛)   𝑁(𝑦)

Proof of Theorem finxpreclem4
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 2onn 7765 . . . . . . . 8 2𝑜 ∈ ω
2 nnon 7113 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ On)
3 2on 7613 . . . . . . . . . . . . . 14 2𝑜 ∈ On
4 oawordeu 7680 . . . . . . . . . . . . . 14 (((2𝑜 ∈ On ∧ 𝑁 ∈ On) ∧ 2𝑜𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
53, 4mpanl1 716 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 2𝑜𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
6 riotasbc 6666 . . . . . . . . . . . . 13 (∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁[(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁)
75, 6syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ On ∧ 2𝑜𝑁) → [(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁)
8 riotaex 6655 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V
9 sbceq1g 4021 . . . . . . . . . . . . . 14 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V → ([(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = 𝑁))
108, 9ax-mp 5 . . . . . . . . . . . . 13 ([(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = 𝑁)
11 csbov2g 6731 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜))
128, 11ax-mp 5 . . . . . . . . . . . . . . 15 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜)
13 csbvarg 4036 . . . . . . . . . . . . . . . . 17 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜 = (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
148, 13ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜 = (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
1514oveq2i 6701 . . . . . . . . . . . . . . 15 (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
1612, 15eqtri 2673 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
1716eqeq1i 2656 . . . . . . . . . . . . 13 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
1810, 17bitri 264 . . . . . . . . . . . 12 ([(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
197, 18sylib 208 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
202, 19sylan 487 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
21 simpl 472 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ ω)
2220, 21eqeltrd 2730 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
23 riotacl 6665 . . . . . . . . . . 11 (∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
24 riotaund 6687 . . . . . . . . . . . 12 (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) = ∅)
25 0elon 5816 . . . . . . . . . . . 12 ∅ ∈ On
2624, 25syl6eqel 2738 . . . . . . . . . . 11 (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
2723, 26pm2.61i 176 . . . . . . . . . 10 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On
28 nnarcl 7741 . . . . . . . . . . . 12 ((2𝑜 ∈ On ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On) → ((2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (2𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)))
293, 28mpan 706 . . . . . . . . . . 11 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On → ((2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (2𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)))
301biantrur 526 . . . . . . . . . . 11 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω ↔ (2𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω))
3129, 30syl6bbr 278 . . . . . . . . . 10 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On → ((2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω))
3227, 31ax-mp 5 . . . . . . . . 9 ((2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
3322, 32sylib 208 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
34 nnacom 7742 . . . . . . . 8 ((2𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜))
351, 33, 34sylancr 696 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜))
36 df-2o 7606 . . . . . . . . 9 2𝑜 = suc 1𝑜
3736oveq2i 6701 . . . . . . . 8 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜)
38 1onn 7764 . . . . . . . . 9 1𝑜 ∈ ω
39 nnasuc 7731 . . . . . . . . 9 (((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω ∧ 1𝑜 ∈ ω) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4033, 38, 39sylancl 695 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4137, 40syl5eq 2697 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4235, 20, 413eqtr3d 2693 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
432adantr 480 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ On)
44 sucidg 5841 . . . . . . . . . . . 12 (1𝑜 ∈ ω → 1𝑜 ∈ suc 1𝑜)
4538, 44ax-mp 5 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
4645, 36eleqtrri 2729 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
47 ssel 3630 . . . . . . . . . 10 (2𝑜𝑁 → (1𝑜 ∈ 2𝑜 → 1𝑜𝑁))
4846, 47mpi 20 . . . . . . . . 9 (2𝑜𝑁 → 1𝑜𝑁)
49 ne0i 3954 . . . . . . . . 9 (1𝑜𝑁𝑁 ≠ ∅)
5048, 49syl 17 . . . . . . . 8 (2𝑜𝑁𝑁 ≠ ∅)
5150adantl 481 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ≠ ∅)
52 nnlim 7120 . . . . . . . 8 (𝑁 ∈ ω → ¬ Lim 𝑁)
5352adantr 480 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ¬ Lim 𝑁)
54 onsucuni3 33345 . . . . . . 7 ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc 𝑁)
5543, 51, 53, 54syl3anc 1366 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = suc 𝑁)
56 nnacom 7742 . . . . . . . 8 (((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω ∧ 1𝑜 ∈ ω) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
5733, 38, 56sylancl 695 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
58 suceq 5828 . . . . . . 7 (((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) → suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
5957, 58syl 17 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
6042, 55, 593eqtr3d 2693 . . . . 5 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → suc 𝑁 = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
61 ordom 7116 . . . . . . . . 9 Ord ω
62 ordelss 5777 . . . . . . . . 9 ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
6361, 62mpan 706 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ⊆ ω)
64 nnfi 8194 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ∈ Fin)
65 nnunifi 8252 . . . . . . . 8 ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → 𝑁 ∈ ω)
6663, 64, 65syl2anc 694 . . . . . . 7 (𝑁 ∈ ω → 𝑁 ∈ ω)
6766adantr 480 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ ω)
68 nnacl 7736 . . . . . . 7 ((1𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
6938, 33, 68sylancr 696 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
70 peano4 7130 . . . . . 6 (( 𝑁 ∈ ω ∧ (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω) → (suc 𝑁 = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ↔ 𝑁 = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7167, 69, 70syl2anc 694 . . . . 5 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (suc 𝑁 = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ↔ 𝑁 = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7260, 71mpbid 222 . . . 4 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
7372fveq2d 6233 . . 3 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7473adantr 480 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7533adantr 480 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
76 finxpreclem4.1 . . . . . . 7 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
7776finxpreclem3 33360 . . . . . 6 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
78 df-1o 7605 . . . . . . . 8 1𝑜 = suc ∅
7978fveq2i 6232 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
80 rdgsuc 7565 . . . . . . . 8 (∅ ∈ On → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)))
8125, 80ax-mp 5 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
82 opex 4962 . . . . . . . . 9 𝑁, 𝑦⟩ ∈ V
8382rdg0 7562 . . . . . . . 8 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦
8483fveq2i 6232 . . . . . . 7 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
8579, 81, 843eqtri 2677 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (𝐹‘⟨𝑁, 𝑦⟩)
8677, 85syl6reqr 2704 . . . . 5 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = ⟨ 𝑁, (1st𝑦)⟩)
8786fveq2d 6233 . . . 4 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩))
88 2on0 7614 . . . . . 6 2𝑜 ≠ ∅
89 nnlim 7120 . . . . . . 7 (2𝑜 ∈ ω → ¬ Lim 2𝑜)
901, 89ax-mp 5 . . . . . 6 ¬ Lim 2𝑜
91 rdgsucuni 33347 . . . . . 6 ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜)))
923, 88, 90, 91mp3an 1464 . . . . 5 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜))
93 1oequni2o 33346 . . . . . . 7 1𝑜 = 2𝑜
9493fveq2i 6232 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜)
9594fveq2i 6232 . . . . 5 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜))
9692, 95eqtr4i 2676 . . . 4 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜))
9778fveq2i 6232 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅)
98 rdgsuc 7565 . . . . . 6 (∅ ∈ On → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)))
9925, 98ax-mp 5 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅))
100 opex 4962 . . . . . . 7 𝑁, (1st𝑦)⟩ ∈ V
101100rdg0 7562 . . . . . 6 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅) = ⟨ 𝑁, (1st𝑦)⟩
102101fveq2i 6232 . . . . 5 (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10397, 99, 1023eqtri 2677 . . . 4 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10487, 96, 1033eqtr4g 2710 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜))
105 1on 7612 . . . 4 1𝑜 ∈ On
106 rdgeqoa 33348 . . . 4 ((2𝑜 ∈ On ∧ 1𝑜 ∈ On ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → ((rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))))
1073, 105, 106mp3an12 1454 . . 3 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω → ((rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))))
10875, 104, 107sylc 65 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
10920fveq2d 6233 . . 3 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
110109adantr 480 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
11174, 108, 1103eqtr2rd 2692 1 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  ∃!wreu 2943  Vcvv 3231  [wsbc 3468  csb 3566  wss 3607  c0 3948  ifcif 4119  cop 4216   cuni 4468   × cxp 5141  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  cfv 5926  crio 6650  (class class class)co 6690  cmpt2 6692  ωcom 7107  1st c1st 7208  reccrdg 7550  1𝑜c1o 7598  2𝑜c2o 7599   +𝑜 coa 7602  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001
This theorem is referenced by:  finxpsuclem  33364
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