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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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