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Theorem finxpreclem4 33542
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 7889 . . . . . . . 8 2𝑜 ∈ ω
2 nnon 7236 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ On)
3 2on 7737 . . . . . . . . . . . . . 14 2𝑜 ∈ On
4 oawordeu 7804 . . . . . . . . . . . . . 14 (((2𝑜 ∈ On ∧ 𝑁 ∈ On) ∧ 2𝑜𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
53, 4mpanl1 718 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 2𝑜𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
6 riotasbc 6789 . . . . . . . . . . . . 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 6778 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V
9 sbceq1g 4131 . . . . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . . 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 4146 . . . . . . . . . . . . . . . . 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 6824 . . . . . . . . . . . . . . 15 (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
1612, 15eqtri 2782 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
1716eqeq1i 2765 . . . . . . . . . . . . 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 489 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
21 simpl 474 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ ω)
2220, 21eqeltrd 2839 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
23 riotacl 6788 . . . . . . . . . . 11 (∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
24 riotaund 6810 . . . . . . . . . . . 12 (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) = ∅)
25 0elon 5939 . . . . . . . . . . . 12 ∅ ∈ On
2624, 25syl6eqel 2847 . . . . . . . . . . 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 7865 . . . . . . . . . . . 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 708 . . . . . . . . . . 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 528 . . . . . . . . . . 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 7866 . . . . . . . 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 698 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜))
36 df-2o 7730 . . . . . . . . 9 2𝑜 = suc 1𝑜
3736oveq2i 6824 . . . . . . . 8 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜)
38 1onn 7888 . . . . . . . . 9 1𝑜 ∈ ω
39 nnasuc 7855 . . . . . . . . 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 697 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4137, 40syl5eq 2806 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4235, 20, 413eqtr3d 2802 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
432adantr 472 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ On)
44 sucidg 5964 . . . . . . . . . . . 12 (1𝑜 ∈ ω → 1𝑜 ∈ suc 1𝑜)
4538, 44ax-mp 5 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
4645, 36eleqtrri 2838 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
47 ssel 3738 . . . . . . . . . 10 (2𝑜𝑁 → (1𝑜 ∈ 2𝑜 → 1𝑜𝑁))
4846, 47mpi 20 . . . . . . . . 9 (2𝑜𝑁 → 1𝑜𝑁)
49 ne0i 4064 . . . . . . . . 9 (1𝑜𝑁𝑁 ≠ ∅)
5048, 49syl 17 . . . . . . . 8 (2𝑜𝑁𝑁 ≠ ∅)
5150adantl 473 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ≠ ∅)
52 nnlim 7243 . . . . . . . 8 (𝑁 ∈ ω → ¬ Lim 𝑁)
5352adantr 472 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ¬ Lim 𝑁)
54 onsucuni3 33526 . . . . . . 7 ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc 𝑁)
5543, 51, 53, 54syl3anc 1477 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = suc 𝑁)
56 nnacom 7866 . . . . . . . 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 697 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
58 suceq 5951 . . . . . . 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 2802 . . . . 5 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → suc 𝑁 = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
61 ordom 7239 . . . . . . . . 9 Ord ω
62 ordelss 5900 . . . . . . . . 9 ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
6361, 62mpan 708 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ⊆ ω)
64 nnfi 8318 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ∈ Fin)
65 nnunifi 8376 . . . . . . . 8 ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → 𝑁 ∈ ω)
6663, 64, 65syl2anc 696 . . . . . . 7 (𝑁 ∈ ω → 𝑁 ∈ ω)
6766adantr 472 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ ω)
68 nnacl 7860 . . . . . . 7 ((1𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
6938, 33, 68sylancr 698 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
70 peano4 7253 . . . . . 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 696 . . . . 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 6356 . . 3 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7473adantr 472 . 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 472 . . 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 33541 . . . . . 6 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
78 df-1o 7729 . . . . . . . 8 1𝑜 = suc ∅
7978fveq2i 6355 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
80 rdgsuc 7689 . . . . . . . 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 5081 . . . . . . . . 9 𝑁, 𝑦⟩ ∈ V
8382rdg0 7686 . . . . . . . 8 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦
8483fveq2i 6355 . . . . . . 7 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
8579, 81, 843eqtri 2786 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (𝐹‘⟨𝑁, 𝑦⟩)
8677, 85syl6reqr 2813 . . . . 5 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = ⟨ 𝑁, (1st𝑦)⟩)
8786fveq2d 6356 . . . 4 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩))
88 2on0 7738 . . . . . 6 2𝑜 ≠ ∅
89 nnlim 7243 . . . . . . 7 (2𝑜 ∈ ω → ¬ Lim 2𝑜)
901, 89ax-mp 5 . . . . . 6 ¬ Lim 2𝑜
91 rdgsucuni 33528 . . . . . 6 ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜)))
923, 88, 90, 91mp3an 1573 . . . . 5 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜))
93 1oequni2o 33527 . . . . . . 7 1𝑜 = 2𝑜
9493fveq2i 6355 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜)
9594fveq2i 6355 . . . . 5 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜))
9692, 95eqtr4i 2785 . . . 4 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜))
9778fveq2i 6355 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅)
98 rdgsuc 7689 . . . . . 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 5081 . . . . . . 7 𝑁, (1st𝑦)⟩ ∈ V
101100rdg0 7686 . . . . . 6 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅) = ⟨ 𝑁, (1st𝑦)⟩
102101fveq2i 6355 . . . . 5 (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10397, 99, 1023eqtri 2786 . . . 4 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10487, 96, 1033eqtr4g 2819 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜))
105 1on 7736 . . . 4 1𝑜 ∈ On
106 rdgeqoa 33529 . . . 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 1563 . . 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 6356 . . 3 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
110109adantr 472 . 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 2801 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 1632  wcel 2139  wne 2932  ∃!wreu 3052  Vcvv 3340  [wsbc 3576  csb 3674  wss 3715  c0 4058  ifcif 4230  cop 4327   cuni 4588   × cxp 5264  Ord word 5883  Oncon0 5884  Lim wlim 5885  suc csuc 5886  cfv 6049  crio 6773  (class class class)co 6813  cmpt2 6815  ωcom 7230  1st c1st 7331  reccrdg 7674  1𝑜c1o 7722  2𝑜c2o 7723   +𝑜 coa 7726  Fincfn 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125
This theorem is referenced by:  finxpsuclem  33545
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