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 = 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 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) ∧ 2𝑜 ⊆ 𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
5	3, 4	mpanl1 691	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 2𝑜 ⊆ 𝑁) → ∃!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)
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ On ∧ 2𝑜 ⊆ 𝑁) → [(℩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))
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 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) / 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	8	csbvargi 4167	. . . . . . . . . . . . . . . 16 ⊢ ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌𝑜 = (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
14	13	oveq2i 6857	. . . . . . . . . . . . . . 15 ⊢ (2𝑜 +𝑜 ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌𝑜) = (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
15	12, 14	eqtri 2787	. . . . . . . . . . . . . 14 ⊢ ⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
16	15	eqeq1i 2770	. . . . . . . . . . . . 13 ⊢ (⦋(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜⦌(2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
17	10, 16	bitri 266	. . . . . . . . . . . 12 ⊢ ([(℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
18	7, 17	sylib 209	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 2𝑜 ⊆ 𝑁) → (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
19	2, 18	sylan 575	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
20		simpl 474	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 ∈ ω)
21	19, 20	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (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
25	23, 24	syl6eqel 2852	. . . . . . . . . . 11 ⊢ (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
26	22, 25	pm2.61i 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)))
28	3, 27	mpan 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)))
29	1	biantrur 526	. . . . . . . . . . 11 ⊢ ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω ↔ (2𝑜 ∈ ω ∧ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω))
30	28, 29	syl6bbr 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))
31	26, 30	ax-mp 5	. . . . . . . . 9 ⊢ ((2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
32	21, 31	sylib 209	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (℩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))
34	1, 32, 33	sylancr 581	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜))
35		df-2o 7769	. . . . . . . . 9 ⊢ 2𝑜 = suc 1𝑜
36	35	oveq2i 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))
39	32, 37, 38	sylancl 580	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜) = suc ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
40	36, 39	syl5eq 2811	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = suc ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
41	34, 19, 40	3eqtr3d 2807	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 = suc ((℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
42	2	adantr 472	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 ∈ On)
43		sucidg 5988	. . . . . . . . . . . 12 ⊢ (1𝑜 ∈ ω → 1𝑜 ∈ suc 1𝑜)
44	37, 43	ax-mp 5	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ suc 1𝑜
45	44, 35	eleqtrri 2843	. . . . . . . . . 10 ⊢ 1𝑜 ∈ 2𝑜
46		ssel 3757	. . . . . . . . . 10 ⊢ (2𝑜 ⊆ 𝑁 → (1𝑜 ∈ 2𝑜 → 1𝑜 ∈ 𝑁))
47	45, 46	mpi 20	. . . . . . . . 9 ⊢ (2𝑜 ⊆ 𝑁 → 1𝑜 ∈ 𝑁)
48	47	ne0d 4088	. . . . . . . 8 ⊢ (2𝑜 ⊆ 𝑁 → 𝑁 ≠ ∅)
49	48	adantl 473	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 𝑁 ≠ ∅)
50		nnlim 7280	. . . . . . . 8 ⊢ (𝑁 ∈ ω → ¬ Lim 𝑁)
51	50	adantr 472	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ¬ Lim 𝑁)
52		onsucuni3 33669	. . . . . . 7 ⊢ ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc ∪ 𝑁)
53	42, 49, 51, 52	syl3anc 1490	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 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)))
55	32, 37, 54	sylancl 580	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ((℩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)))
57	55, 56	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)))
58	41, 53, 57	3eqtr3d 2807	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → 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)
61	59, 60	mpan 681	. . . . . . . 8 ⊢ (𝑁 ∈ ω → 𝑁 ⊆ ω)
62		nnfi 8364	. . . . . . . 8 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
63		nnunifi 8422	. . . . . . . 8 ⊢ ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → ∪ 𝑁 ∈ ω)
64	61, 62, 63	syl2anc 579	. . . . . . 7 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
65	64	adantr 472	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ∪ 𝑁 ∈ ω)
66		nnacl 7900	. . . . . . 7 ⊢ ((1𝑜 ∈ ω ∧ (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
67	37, 32, 66	sylancr 581	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (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))))
69	65, 67, 68	syl2anc 579	. . . . 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))))
70	58, 69	mpbid 223	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → ∪ 𝑁 = (1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
71	70	fveq2d 6383	. . 3 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∪ 𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘(1𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
72	71	adantr 472	. 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))))
73	32	adantr 472	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
74		finxpreclem4.1	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
75	74	finxpreclem3 33684	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
76		df-1o 7768	. . . . . . . 8 ⊢ 1𝑜 = suc ∅
77	76	fveq2i 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⟩)‘∅)))
79	24, 78	ax-mp 5	. . . . . . 7 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
80		opex 5090	. . . . . . . . 9 ⊢ ⟨𝑁, 𝑦⟩ ∈ V
81	80	rdg0 7725	. . . . . . . 8 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦⟩
82	81	fveq2i 6382	. . . . . . 7 ⊢ (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
83	77, 79, 82	3eqtri 2791	. . . . . 6 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (𝐹‘⟨𝑁, 𝑦⟩)
84	75, 83	syl6reqr 2818	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = ⟨∪ 𝑁, (1st ‘𝑦)⟩)
85	84	fveq2d 6383	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩))
86		2on0 7778	. . . . . 6 ⊢ 2𝑜 ≠ ∅
87		nnlim 7280	. . . . . . 7 ⊢ (2𝑜 ∈ ω → ¬ Lim 2𝑜)
88	1, 87	ax-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)))
90	3, 86, 88, 89	mp3an 1585	. . . . 5 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2𝑜))
91		1oequni2o 33670	. . . . . . 7 ⊢ 1𝑜 = ∪ 2𝑜
92	91	fveq2i 6382	. . . . . 6 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2𝑜)
93	92	fveq2i 6382	. . . . 5 ⊢ (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2𝑜))
94	90, 93	eqtr4i 2790	. . . 4 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜))
95	76	fveq2i 6382	. . . . 5 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1𝑜) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘suc ∅)
96		rdgsuc 7728	. . . . . 6 ⊢ (∅ ∈ On → (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅)))
97	24, 96	ax-mp 5	. . . . 5 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅))
98		opex 5090	. . . . . . 7 ⊢ ⟨∪ 𝑁, (1st ‘𝑦)⟩ ∈ V
99	98	rdg0 7725	. . . . . 6 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅) = ⟨∪ 𝑁, (1st ‘𝑦)⟩
100	99	fveq2i 6382	. . . . 5 ⊢ (𝐹‘(rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅)) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩)
101	95, 97, 100	3eqtri 2791	. . . 4 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1𝑜) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩)
102	85, 94, 101	3eqtr4g 2824	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘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, (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)))))
105	3, 103, 104	mp3an12 1575	. . 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)))))
106	73, 102, 105	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))))
107	19	fveq2d 6383	. . 3 ⊢ ((𝑁 ∈ ω ∧ 2𝑜 ⊆ 𝑁) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (℩𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
108	107	adantr 472	. 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))
109	72, 106, 108	3eqtr2rd 2806	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 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  1^st c1st 7368  reccrdg 7713  1_𝑜c1o 7761  2_𝑜c2o 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