Theorem finxpreclem4 33231

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem4.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

Assertion

Ref	Expression
finxpreclem4	⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∪ 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.

Step	Hyp	Ref	Expression
1		2onn 7720	. . . . . . . 8 ⊢ 2𝑜 ∈ ω
2		nnon 7071	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
3		2on 7568	. . . . . . . . . . . . . 14 ⊢ 2𝑜 ∈ On
4		oawordeu 7635	. . . . . . . . . . . . . 14 ⊢ (((2𝑜 ∈ On ∧ 𝑁 ∈ On) ∧ 2𝑜 ⊆ 𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
5	3, 4	mpanl1 716	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 2𝑜 ⊆ 𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
6		riotasbc 6626	. . . . . . . . . . . . 13 ⊢ (∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → [(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁)
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ On ∧ 2𝑜 ⊆ 𝑁) → [(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁)
8		riotaex 6615	. . . . . . . . . . . . . 14 ⊢ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V
9		sbceq1g 3988	. . . . . . . . . . . . . 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))
10	8, 9	ax-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 6691	. . . . . . . . . . . . . . . 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) / 𝑜⦌𝑜))
12	8, 11	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌𝑜)
13		csbvarg 4003	. . . . . . . . . . . . . . . . 17 ⊢ ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V → ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌𝑜 = (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
14	8, 13	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌𝑜 = (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
15	14	oveq2i 6661	. . . . . . . . . . . . . . 15 ⊢ (2𝑜 +𝑜 ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌𝑜) = (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
16	12, 15	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
17	16	eqeq1i 2627	. . . . . . . . . . . . 13 ⊢ (⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌(2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
18	10, 17	bitri 264	. . . . . . . . . . . 12 ⊢ ([(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
19	7, 18	sylib 208	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 2𝑜 ⊆ 𝑁) → (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
20	2, 19	sylan 488	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
21		simpl 473	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 ∈ ω)
22	20, 21	eqeltrd 2701	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
23		riotacl 6625	. . . . . . . . . . 11 ⊢ (∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
24		riotaund 6647	. . . . . . . . . . . 12 ⊢ (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) = ∅)
25		0elon 5778	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
26	24, 25	syl6eqel 2709	. . . . . . . . . . 11 ⊢ (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
27	23, 26	pm2.61i 176	. . . . . . . . . 10 ⊢ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On
28		nnarcl 7696	. . . . . . . . . . . 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)))
29	3, 28	mpan 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)))
30	1	biantrur 527	. . . . . . . . . . 11 ⊢ ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω ↔ (2𝑜 ∈ ω ∧ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω))
31	29, 30	syl6bbr 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))
32	27, 31	ax-mp 5	. . . . . . . . 9 ⊢ ((2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
33	22, 32	sylib 208	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
34		nnacom 7697	. . . . . . . 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))
35	1, 33, 34	sylancr 695	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜))
36		df-2o 7561	. . . . . . . . 9 ⊢ 2𝑜 = suc 1𝑜
37	36	oveq2i 6661	. . . . . . . 8 ⊢ ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜)
38		1onn 7719	. . . . . . . . 9 ⊢ 1𝑜 ∈ ω
39		nnasuc 7686	. . . . . . . . 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))
40	33, 38, 39	sylancl 694	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜) = suc ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
41	37, 40	syl5eq 2668	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = suc ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
42	35, 20, 41	3eqtr3d 2664	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 = suc ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
43	2	adantr 481	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 ∈ On)
44		sucidg 5803	. . . . . . . . . . . 12 ⊢ (1𝑜 ∈ ω → 1𝑜 ∈ suc 1𝑜)
45	38, 44	ax-mp 5	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ suc 1𝑜
46	45, 36	eleqtrri 2700	. . . . . . . . . 10 ⊢ 1𝑜 ∈ 2𝑜
47		ssel 3597	. . . . . . . . . 10 ⊢ (2𝑜 ⊆ 𝑁 → (1𝑜 ∈ 2𝑜 → 1𝑜 ∈ 𝑁))
48	46, 47	mpi 20	. . . . . . . . 9 ⊢ (2𝑜 ⊆ 𝑁 → 1𝑜 ∈ 𝑁)
49		ne0i 3921	. . . . . . . . 9 ⊢ (1𝑜 ∈ 𝑁 → 𝑁 ≠ ∅)
50	48, 49	syl 17	. . . . . . . 8 ⊢ (2𝑜 ⊆ 𝑁 → 𝑁 ≠ ∅)
51	50	adantl 482	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 ≠ ∅)
52		nnlim 7078	. . . . . . . 8 ⊢ (𝑁 ∈ ω → ¬ Lim 𝑁)
53	52	adantr 481	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ¬ Lim 𝑁)
54		onsucuni3 33215	. . . . . . 7 ⊢ ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc ∪ 𝑁)
55	43, 51, 53, 54	syl3anc 1326	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 = suc ∪ 𝑁)
56		nnacom 7697	. . . . . . . 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)))
57	33, 38, 56	sylancl 694	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
58		suceq 5790	. . . . . . 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)))
59	57, 58	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → suc ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = suc (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
60	42, 55, 59	3eqtr3d 2664	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → suc ∪ 𝑁 = suc (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
61		ordom 7074	. . . . . . . . 9 ⊢ Ord ω
62		ordelss 5739	. . . . . . . . 9 ⊢ ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
63	61, 62	mpan 706	. . . . . . . 8 ⊢ (𝑁 ∈ ω → 𝑁 ⊆ ω)
64		nnfi 8153	. . . . . . . 8 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
65		nnunifi 8211	. . . . . . . 8 ⊢ ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → ∪ 𝑁 ∈ ω)
66	63, 64, 65	syl2anc 693	. . . . . . 7 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
67	66	adantr 481	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ∪ 𝑁 ∈ ω)
68		nnacl 7691	. . . . . . 7 ⊢ ((1𝑜 ∈ ω ∧ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
69	38, 33, 68	sylancr 695	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
70		peano4 7088	. . . . . 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))))
71	67, 69, 70	syl2anc 693	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (suc ∪ 𝑁 = suc (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ↔ ∪ 𝑁 = (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
72	60, 71	mpbid 222	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ∪ 𝑁 = (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
73	72	fveq2d 6195	. . 3 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∪ 𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘(1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
74	73	adantr 481	. 2 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∪ 𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘(1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
75	33	adantr 481	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
76		finxpreclem4.1	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
77	76	finxpreclem3 33230	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
78		df-1o 7560	. . . . . . . 8 ⊢ 1𝑜 = suc ∅
79	78	fveq2i 6194	. . . . . . 7 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
80		rdgsuc 7520	. . . . . . . 8 ⊢ (∅ ∈ On → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)))
81	25, 80	ax-mp 5	. . . . . . 7 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
82		opex 4932	. . . . . . . . 9 ⊢ ⟨𝑁, 𝑦⟩ ∈ V
83	82	rdg0 7517	. . . . . . . 8 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦⟩
84	83	fveq2i 6194	. . . . . . 7 ⊢ (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
85	79, 81, 84	3eqtri 2648	. . . . . 6 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (𝐹‘⟨𝑁, 𝑦⟩)
86	77, 85	syl6reqr 2675	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = ⟨∪ 𝑁, (1st ‘𝑦)⟩)
87	86	fveq2d 6195	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩))
88		2on0 7569	. . . . . 6 ⊢ 2𝑜 ≠ ∅
89		nnlim 7078	. . . . . . 7 ⊢ (2𝑜 ∈ ω → ¬ Lim 2𝑜)
90	1, 89	ax-mp 5	. . . . . 6 ⊢ ¬ Lim 2𝑜
91		rdgsucuni 33217	. . . . . 6 ⊢ ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2𝑜)))
92	3, 88, 90, 91	mp3an 1424	. . . . 5 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2𝑜))
93		1oequni2o 33216	. . . . . . 7 ⊢ 1𝑜 = ∪ 2𝑜
94	93	fveq2i 6194	. . . . . 6 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2𝑜)
95	94	fveq2i 6194	. . . . 5 ⊢ (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2𝑜))
96	92, 95	eqtr4i 2647	. . . 4 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜))
97	78	fveq2i 6194	. . . . 5 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1𝑜) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘suc ∅)
98		rdgsuc 7520	. . . . . 6 ⊢ (∅ ∈ On → (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅)))
99	25, 98	ax-mp 5	. . . . 5 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅))
100		opex 4932	. . . . . . 7 ⊢ ⟨∪ 𝑁, (1st ‘𝑦)⟩ ∈ V
101	100	rdg0 7517	. . . . . 6 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅) = ⟨∪ 𝑁, (1st ‘𝑦)⟩
102	101	fveq2i 6194	. . . . 5 ⊢ (𝐹‘(rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅)) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩)
103	97, 99, 102	3eqtri 2648	. . . 4 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1𝑜) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩)
104	87, 96, 103	3eqtr4g 2681	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1𝑜))
105		1on 7567	. . . 4 ⊢ 1𝑜 ∈ On
106		rdgeqoa 33218	. . . 4 ⊢ ((2𝑜 ∈ On ∧ 1𝑜 ∈ On ∧ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → ((rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘(1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))))
107	3, 105, 106	mp3an12 1414	. . 3 ⊢ ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω → ((rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘(1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))))
108	75, 104, 107	sylc 65	. 2 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘(1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
109	20	fveq2d 6195	. . 3 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
110	109	adantr 481	. 2 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
111	74, 108, 110	3eqtr2rd 2663	1 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∪ 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃!wreu 2914  Vcvv 3200  [wsbc 3435  ⦋csb 3533   ⊆ wss 3574  ∅c0 3915  ifcif 4086  ⟨cop 4183  ∪ cuni 4436   × cxp 5112  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  1^st c1st 7166  reccrdg 7505  1_𝑜c1o 7553  2_𝑜c2o 7554   +_𝑜 coa 7557  Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  finxpsuclem  33234