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Theorem finxpreclem4 33685
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 7929 . . . . . . . 8 2𝑜 ∈ ω
2 nnon 7273 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ On)
3 2on 7777 . . . . . . . . . . . . . 14 2𝑜 ∈ On
4 oawordeu 7844 . . . . . . . . . . . . . 14 (((2𝑜 ∈ On ∧ 𝑁 ∈ On) ∧ 2𝑜𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
53, 4mpanl1 691 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 2𝑜𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
6 riotasbc 6822 . . . . . . . . . . . . 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 6811 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V
9 sbceq1g 4151 . . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . 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)
138csbvargi 4167 . . . . . . . . . . . . . . . 16 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜 = (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
1413oveq2i 6857 . . . . . . . . . . . . . . 15 (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
1512, 14eqtri 2787 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
1615eqeq1i 2770 . . . . . . . . . . . . 13 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
1710, 16bitri 266 . . . . . . . . . . . 12 ([(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
187, 17sylib 209 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
192, 18sylan 575 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
20 simpl 474 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ ω)
2119, 20eqeltrd 2844 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
22 riotacl 6821 . . . . . . . . . . 11 (∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
23 riotaund 6843 . . . . . . . . . . . 12 (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) = ∅)
24 0elon 5963 . . . . . . . . . . . 12 ∅ ∈ On
2523, 24syl6eqel 2852 . . . . . . . . . . 11 (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
2622, 25pm2.61i 176 . . . . . . . . . 10 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On
27 nnarcl 7905 . . . . . . . . . . . 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)))
283, 27mpan 681 . . . . . . . . . . 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)))
291biantrur 526 . . . . . . . . . . 11 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω ↔ (2𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω))
3028, 29syl6bbr 280 . . . . . . . . . 10 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On → ((2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω))
3126, 30ax-mp 5 . . . . . . . . 9 ((2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
3221, 31sylib 209 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
33 nnacom 7906 . . . . . . . 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))
341, 32, 33sylancr 581 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜))
35 df-2o 7769 . . . . . . . . 9 2𝑜 = suc 1𝑜
3635oveq2i 6857 . . . . . . . 8 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜)
37 1onn 7928 . . . . . . . . 9 1𝑜 ∈ ω
38 nnasuc 7895 . . . . . . . . 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))
3932, 37, 38sylancl 580 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4036, 39syl5eq 2811 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4134, 19, 403eqtr3d 2807 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
422adantr 472 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ On)
43 sucidg 5988 . . . . . . . . . . . 12 (1𝑜 ∈ ω → 1𝑜 ∈ suc 1𝑜)
4437, 43ax-mp 5 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
4544, 35eleqtrri 2843 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
46 ssel 3757 . . . . . . . . . 10 (2𝑜𝑁 → (1𝑜 ∈ 2𝑜 → 1𝑜𝑁))
4745, 46mpi 20 . . . . . . . . 9 (2𝑜𝑁 → 1𝑜𝑁)
4847ne0d 4088 . . . . . . . 8 (2𝑜𝑁𝑁 ≠ ∅)
4948adantl 473 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ≠ ∅)
50 nnlim 7280 . . . . . . . 8 (𝑁 ∈ ω → ¬ Lim 𝑁)
5150adantr 472 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ¬ Lim 𝑁)
52 onsucuni3 33669 . . . . . . 7 ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc 𝑁)
5342, 49, 51, 52syl3anc 1490 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = suc 𝑁)
54 nnacom 7906 . . . . . . . 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)))
5532, 37, 54sylancl 580 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
56 suceq 5975 . . . . . . 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)))
5755, 56syl 17 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
5841, 53, 573eqtr3d 2807 . . . . 5 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → suc 𝑁 = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
59 ordom 7276 . . . . . . . . 9 Ord ω
60 ordelss 5926 . . . . . . . . 9 ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
6159, 60mpan 681 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ⊆ ω)
62 nnfi 8364 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ∈ Fin)
63 nnunifi 8422 . . . . . . . 8 ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → 𝑁 ∈ ω)
6461, 62, 63syl2anc 579 . . . . . . 7 (𝑁 ∈ ω → 𝑁 ∈ ω)
6564adantr 472 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ ω)
66 nnacl 7900 . . . . . . 7 ((1𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
6737, 32, 66sylancr 581 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
68 peano4 7290 . . . . . 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))))
6965, 67, 68syl2anc 579 . . . . 5 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (suc 𝑁 = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ↔ 𝑁 = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7058, 69mpbid 223 . . . 4 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
7170fveq2d 6383 . . 3 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7271adantr 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))))
7332adantr 472 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
74 finxpreclem4.1 . . . . . . 7 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
7574finxpreclem3 33684 . . . . . 6 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
76 df-1o 7768 . . . . . . . 8 1𝑜 = suc ∅
7776fveq2i 6382 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
78 rdgsuc 7728 . . . . . . . 8 (∅ ∈ On → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)))
7924, 78ax-mp 5 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
80 opex 5090 . . . . . . . . 9 𝑁, 𝑦⟩ ∈ V
8180rdg0 7725 . . . . . . . 8 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦
8281fveq2i 6382 . . . . . . 7 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
8377, 79, 823eqtri 2791 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (𝐹‘⟨𝑁, 𝑦⟩)
8475, 83syl6reqr 2818 . . . . 5 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = ⟨ 𝑁, (1st𝑦)⟩)
8584fveq2d 6383 . . . 4 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩))
86 2on0 7778 . . . . . 6 2𝑜 ≠ ∅
87 nnlim 7280 . . . . . . 7 (2𝑜 ∈ ω → ¬ Lim 2𝑜)
881, 87ax-mp 5 . . . . . 6 ¬ Lim 2𝑜
89 rdgsucuni 33671 . . . . . 6 ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜)))
903, 86, 88, 89mp3an 1585 . . . . 5 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜))
91 1oequni2o 33670 . . . . . . 7 1𝑜 = 2𝑜
9291fveq2i 6382 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜)
9392fveq2i 6382 . . . . 5 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜))
9490, 93eqtr4i 2790 . . . 4 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜))
9576fveq2i 6382 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅)
96 rdgsuc 7728 . . . . . 6 (∅ ∈ On → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)))
9724, 96ax-mp 5 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅))
98 opex 5090 . . . . . . 7 𝑁, (1st𝑦)⟩ ∈ V
9998rdg0 7725 . . . . . 6 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅) = ⟨ 𝑁, (1st𝑦)⟩
10099fveq2i 6382 . . . . 5 (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10195, 97, 1003eqtri 2791 . . . 4 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10285, 94, 1013eqtr4g 2824 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜))
103 1on 7775 . . . 4 1𝑜 ∈ On
104 rdgeqoa 33672 . . . 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)))))
1053, 103, 104mp3an12 1575 . . 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)))))
10673, 102, 105sylc 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))))
10719fveq2d 6383 . . 3 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
108107adantr 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))
10972, 106, 1083eqtr2rd 2806 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 197  wa 384   = wceq 1652  wcel 2155  wne 2937  ∃!wreu 3057  Vcvv 3350  [wsbc 3598  csb 3693  wss 3734  c0 4081  ifcif 4245  cop 4342   cuni 4596   × cxp 5277  Ord word 5909  Oncon0 5910  Lim wlim 5911  suc csuc 5912  cfv 6070  crio 6806  (class class class)co 6846  cmpt2 6848  ωcom 7267  1st c1st 7368  reccrdg 7713  1𝑜c1o 7761  2𝑜c2o 7762   +𝑜 coa 7765  Fincfn 8164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168
This theorem is referenced by:  finxpsuclem  33688
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