Theorem finxpreclem4 37927

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 8627	. . . . . . . 8 ⊢ 2o ∈ ω
2		nnon 7867	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
3		2on 8466	. . . . . . . . . . . . . 14 ⊢ 2o ∈ On
4		oawordeu 8539	. . . . . . . . . . . . . 14 ⊢ (((2o ∈ On ∧ 𝑁 ∈ On) ∧ 2o ⊆ 𝑁) → ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁)
5	3, 4	mpanl1 712	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 2o ⊆ 𝑁) → ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁)
6		riotasbc 7386	. . . . . . . . . . . . 13 ⊢ (∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → [(℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁)
7	5, 6	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ On ∧ 2o ⊆ 𝑁) → [(℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁)
8		riotaex 7372	. . . . . . . . . . . . . 14 ⊢ (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ V
9		sbceq1g 4388	. . . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . 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 4406	. . . . . . . . . . . . . . . 16 ⊢ ⦋(℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜⦌𝑜 = (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)
14	13	oveq2i 7422	. . . . . . . . . . . . . . 15 ⊢ (2o +o ⦋(℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜⦌𝑜) = (2o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁))
15	12, 14	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ ⦋(℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜⦌(2o +o 𝑜) = (2o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁))
16	15	eqeq1i 2774	. . . . . . . . . . . . 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 278	. . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 2o ⊆ 𝑁) → (2o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
19	2, 18	sylan 591	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → (2o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
20		simpl 487	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → 𝑁 ∈ ω)
21	19, 20	eqeltrd 2869	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → (2o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
22		riotacl 7385	. . . . . . . . . . 11 ⊢ (∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On)
23		riotaund 7407	. . . . . . . . . . . 12 ⊢ (¬ ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) = ∅)
24		0elon 6417	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
25	23, 24	eqeltrdi 2877	. . . . . . . . . . 11 ⊢ (¬ ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On)
26	22, 25	pm2.61i 184	. . . . . . . . . 10 ⊢ (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On
27		nnarcl 8601	. . . . . . . . . . . 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 702	. . . . . . . . . . 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 539	. . . . . . . . . . 11 ⊢ ((℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω ↔ (2o ∈ ω ∧ (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω))
30	28, 29	bitr4di 292	. . . . . . . . . 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 221	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
33		nnacom 8602	. . . . . . . 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 598	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → (2o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = ((℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o))
35		df-2o 8453	. . . . . . . . 9 ⊢ 2o = suc 1o
36	35	oveq2i 7422	. . . . . . . 8 ⊢ ((℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o) = ((℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o suc 1o)
37		1onn 8625	. . . . . . . . 9 ⊢ 1o ∈ ω
38		nnasuc 8591	. . . . . . . . 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 597	. . . . . . . 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	eqtrid 2816	. . . . . . 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 2812	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → 𝑁 = suc ((℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
42	2	adantr 485	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → 𝑁 ∈ On)
43		sucidg 6445	. . . . . . . . . . . 12 ⊢ (1o ∈ ω → 1o ∈ suc 1o)
44	37, 43	ax-mp 5	. . . . . . . . . . 11 ⊢ 1o ∈ suc 1o
45	44, 35	eleqtrri 2868	. . . . . . . . . 10 ⊢ 1o ∈ 2o
46		ssel 3939	. . . . . . . . . 10 ⊢ (2o ⊆ 𝑁 → (1o ∈ 2o → 1o ∈ 𝑁))
47	45, 46	mpi 21	. . . . . . . . 9 ⊢ (2o ⊆ 𝑁 → 1o ∈ 𝑁)
48	47	ne0d 4303	. . . . . . . 8 ⊢ (2o ⊆ 𝑁 → 𝑁 ≠ ∅)
49	48	adantl 486	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → 𝑁 ≠ ∅)
50		nnlim 7875	. . . . . . . 8 ⊢ (𝑁 ∈ ω → ¬ Lim 𝑁)
51	50	adantr 485	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → ¬ Lim 𝑁)
52		onsucuni3 37900	. . . . . . 7 ⊢ ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc ∪ 𝑁)
53	42, 49, 51, 52	syl3anc 1396	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → 𝑁 = suc ∪ 𝑁)
54		nnacom 8602	. . . . . . . 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 597	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → ((℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = (1o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
56		suceq 6430	. . . . . . 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 18	. . . . . 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 2812	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → suc ∪ 𝑁 = suc (1o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
59		ordom 7871	. . . . . . . . 9 ⊢ Ord ω
60		ordelss 6377	. . . . . . . . 9 ⊢ ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
61	59, 60	mpan 702	. . . . . . . 8 ⊢ (𝑁 ∈ ω → 𝑁 ⊆ ω)
62		nnfi 9151	. . . . . . . 8 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
63		nnunifi 9250	. . . . . . . 8 ⊢ ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → ∪ 𝑁 ∈ ω)
64	61, 62, 63	syl2anc 595	. . . . . . 7 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
65	64	adantr 485	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → ∪ 𝑁 ∈ ω)
66		nnacl 8596	. . . . . . 7 ⊢ ((1o ∈ ω ∧ (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω) → (1o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
67	37, 32, 66	sylancr 598	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → (1o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
68		peano4 7888	. . . . . 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 595	. . . . 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 235	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) → ∪ 𝑁 = (1o +o (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
71	70	fveq2d 6886	. . 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 485	. 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 485	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (℩𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
74		df-1o 8452	. . . . . . . 8 ⊢ 1o = suc ∅
75	74	fveq2i 6885	. . . . . . 7 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
76		rdgsuc 8410	. . . . . . . 8 ⊢ (∅ ∈ On → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)))
77	24, 76	ax-mp 5	. . . . . . 7 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
78		opex 5446	. . . . . . . . 9 ⊢ ⟨𝑁, 𝑦⟩ ∈ V
79	78	rdg0 8407	. . . . . . . 8 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦⟩
80	79	fveq2i 6885	. . . . . . 7 ⊢ (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
81	75, 77, 80	3eqtri 2796	. . . . . 6 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (𝐹‘⟨𝑁, 𝑦⟩)
82		finxpreclem4.1	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
83	82	finxpreclem3 37926	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
84	81, 83	eqtr4id 2823	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = ⟨∪ 𝑁, (1st ‘𝑦)⟩)
85	84	fveq2d 6886	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o)) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩))
86		2on0 8467	. . . . . 6 ⊢ 2o ≠ ∅
87		nnlim 7875	. . . . . . 7 ⊢ (2o ∈ ω → ¬ Lim 2o)
88	1, 87	ax-mp 5	. . . . . 6 ⊢ ¬ Lim 2o
89		rdgsucuni 37902	. . . . . 6 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2o)))
90	3, 86, 88, 89	mp3an 1487	. . . . 5 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2o))
91		1oequni2o 37901	. . . . . . 7 ⊢ 1o = ∪ 2o
92	91	fveq2i 6885	. . . . . 6 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2o)
93	92	fveq2i 6885	. . . . 5 ⊢ (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∪ 2o))
94	90, 93	eqtr4i 2795	. . . 4 ⊢ (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o))
95	74	fveq2i 6885	. . . . 5 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1o) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘suc ∅)
96		rdgsuc 8410	. . . . . 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 5446	. . . . . . 7 ⊢ ⟨∪ 𝑁, (1st ‘𝑦)⟩ ∈ V
99	98	rdg0 8407	. . . . . 6 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅) = ⟨∪ 𝑁, (1st ‘𝑦)⟩
100	99	fveq2i 6885	. . . . 5 ⊢ (𝐹‘(rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∅)) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩)
101	95, 97, 100	3eqtri 2796	. . . 4 ⊢ (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1o) = (𝐹‘⟨∪ 𝑁, (1st ‘𝑦)⟩)
102	85, 94, 101	3eqtr4g 2829	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘1o))
103		1on 8465	. . . 4 ⊢ 1o ∈ On
104		rdgeqoa 37903	. . . 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 1477	. . 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 66	. 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 6886	. . 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 485	. 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 2811	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 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃!wreu 3374  Vcvv 3463  [wsbc 3753  ⦋csb 3861   ⊆ wss 3913  ∅c0 4294  ifcif 4492  ⟨cop 4600  ∪ cuni 4876   × cxp 5660  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  ‘cfv 6537  ℩crio 7367  (class class class)co 7411   ∈ cmpo 7413  ωcom 7861  1^st c1st 7983  reccrdg 8395  1_oc1o 8445  2_oc2o 8446   +_o coa 8449  Fincfn 8942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-en 8943  df-fin 8946

This theorem is referenced by:  finxpsuclem  37930