Theorem finxpreclem6 37601

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem5.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

Assertion

Ref	Expression
finxpreclem6	⊢ ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))

Distinct variable groups:   𝑥,𝑛,𝑁   𝑈,𝑛,𝑥

Allowed substitution hints:   𝐹(𝑥,𝑛)

Proof of Theorem finxpreclem6

Dummy variables 𝑚 𝑜 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2824	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ ω ↔ 𝑁 ∈ ω))
2		eleq2 2825	. . . . 5 ⊢ (𝑛 = 𝑁 → (1o ∈ 𝑛 ↔ 1o ∈ 𝑁))
3	1, 2	anbi12d 632	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ↔ (𝑁 ∈ ω ∧ 1o ∈ 𝑁)))
4		anass 468	. . . . . . . . 9 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
5		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))
6		finxpreclem5.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
7		nfmpo2 7439	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
8	6, 7	nfcxfr 2896	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐹
9		nfcv 2898	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥⟨𝑛, 𝑦⟩
10	8, 9	nfrdg 8345	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥rec(𝐹, ⟨𝑛, 𝑦⟩)
11		nfcv 2898	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑛
12	10, 11	nffv 6844	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
13	12	nfeq2 2916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
14	13	nfn 1858	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
15	5, 14	nfim 1897	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
16		eleq1 2824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (V × 𝑈) ↔ 𝑦 ∈ (V × 𝑈)))
17	16	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (V × 𝑈) ↔ ¬ 𝑦 ∈ (V × 𝑈)))
18	17	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
19	18	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) ↔ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))))
20		opeq2 4830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩)
21		rdgeq2 8343	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑥⟩) = rec(𝐹, ⟨𝑛, 𝑦⟩))
22	20, 21	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → rec(𝐹, ⟨𝑛, 𝑥⟩) = rec(𝐹, ⟨𝑛, 𝑦⟩))
23	22	fveq1d 6836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
24	23	eqeq2d 2747	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
25	24	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
26	19, 25	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛)) ↔ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
27		anass 468	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))))
28		vex 3444	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑛 ∈ V
29		fveqeq2 6843	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ∅ → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩))
30		fveqeq2 6843	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩))
31		fveqeq2 6843	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = suc 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
32		opex 5412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
33	32	rdg0 8352	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩
34	33	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩)
35		nnon 7814	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑜 ∈ ω → 𝑜 ∈ On)
36		rdgsuc 8355	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑜 ∈ On → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)))
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑜 ∈ ω → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)))
38		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)) = (𝐹‘⟨𝑛, 𝑥⟩))
39	37, 38	sylan9eq 2791	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘⟨𝑛, 𝑥⟩))
40	6	finxpreclem5 37600	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))
41	40	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
42	39, 41	sylan9eq 2791	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) ∧ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩)
43	42	expl 457	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑜 ∈ ω → (((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ ∧ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
44	43	expcomd 416	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑜 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩)))
45	29, 30, 31, 34, 44	finds2 7840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
46		eleq1 2824	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ ω ↔ 𝑚 ∈ ω))
47		fveqeq2 6843	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
48	47	imbi2d 340	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → ((((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩) ↔ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩)))
49	46, 48	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → ((𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)) ↔ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))))
50	45, 49	mpbiri 258	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
51	50	equcoms 2021	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
52	28, 51	vtocle 3512	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
53	27, 52	biimtrrid 243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ω → ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
54	53	anabsi5 669	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)
55		vex 3444	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
56	28, 55	opnzi 5422	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑛, 𝑥⟩ ≠ ∅
57	56	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ⟨𝑛, 𝑥⟩ ≠ ∅)
58	54, 57	eqnetrd 2999	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ≠ ∅)
59	58	necomd 2987	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ∅ ≠ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
60	59	neneqd 2937	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
61	15, 26, 60	chvarfv 2247	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
62	61	intnand 488	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
63	62	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
64		opeq1 4829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑁 → ⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩)
65		rdgeq2 8343	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩))
66	64, 65	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑁 → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩))
67		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
68	66, 67	fveq12d 6841	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑁 → (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
69	68	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑁 → (∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁)))
70	1, 69	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))))
71	70	abbidv 2802	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))})
72	6	dffinxpf 37590	. . . . . . . . . . . . . . 15 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}
73	71, 72	eqtr4di 2789	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = (𝑈↑↑𝑁))
74	73	eleq2d 2822	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ 𝑦 ∈ (𝑈↑↑𝑁)))
75		abid 2718	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
76	74, 75	bitr3di 286	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
77	76	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
78	63, 77	mtbird 325	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))
79	78	ex 412	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
80	4, 79	biimtrid 242	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
81	80	expdimp 452	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈ 𝑛)) → (¬ 𝑦 ∈ (V × 𝑈) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
82	81	con4d 115	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈ 𝑛)) → (𝑦 ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (V × 𝑈)))
83	82	ssrdv 3939	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈ 𝑛)) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
84	83	ex 412	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
85	3, 84	sylbird 260	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
86	85	vtocleg 3510	. 2 ⊢ (𝑁 ∈ ω → ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
87	86	anabsi5 669	1 ⊢ ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714   ≠ wne 2932  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  ifcif 4479  ⟨cop 4586  ∪ cuni 4863   × cxp 5622  Oncon0 6317  suc csuc 6319  ‘cfv 6492   ∈ cmpo 7360  ωcom 7808  1^st c1st 7931  reccrdg 8340  1_oc1o 8390  ↑↑cfinxp 37588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-finxp 37589

This theorem is referenced by:  finxpsuclem  37602