Theorem finxpreclem6 36277

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 2822	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ ω ↔ 𝑁 ∈ ω))
2		eleq2 2823	. . . . 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 470	. . . . . . . . 9 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
5		nfv 1918	. . . . . . . . . . . . . . 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 7490	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
8	6, 7	nfcxfr 2902	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐹
9		nfcv 2904	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥⟨𝑛, 𝑦⟩
10	8, 9	nfrdg 8414	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥rec(𝐹, ⟨𝑛, 𝑦⟩)
11		nfcv 2904	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑛
12	10, 11	nffv 6902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
13	12	nfeq2 2921	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
14	13	nfn 1861	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
15	5, 14	nfim 1900	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
16		eleq1 2822	. . . . . . . . . . . . . . . . . 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 4875	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩)
21		rdgeq2 8412	. . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
24	23	eqeq2d 2744	. . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))))
28		vex 3479	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑛 ∈ V
29		fveqeq2 6901	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ∅ → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩))
30		fveqeq2 6901	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩))
31		fveqeq2 6901	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = suc 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
32		opex 5465	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
33	32	rdg0 8421	. . . . . . . . . . . . . . . . . . . . . . . 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 7861	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑜 ∈ ω → 𝑜 ∈ On)
36		rdgsuc 8424	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)) = (𝐹‘⟨𝑛, 𝑥⟩))
39	37, 38	sylan9eq 2793	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘⟨𝑛, 𝑥⟩))
40	6	finxpreclem5 36276	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))
41	40	imp 408	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
42	39, 41	sylan9eq 2793	. . . . . . . . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . . . . . 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 7891	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
46		eleq1 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ ω ↔ 𝑚 ∈ ω))
47		fveqeq2 6901	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
48	47	imbi2d 341	. . . . . . . . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . . . . . . . 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 2024	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
52	28, 51	vtocle 3576	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
53	27, 52	biimtrrid 242	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ω → ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
54	53	anabsi5 668	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)
55		vex 3479	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
56	28, 55	opnzi 5475	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑛, 𝑥⟩ ≠ ∅
57	56	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ⟨𝑛, 𝑥⟩ ≠ ∅)
58	54, 57	eqnetrd 3009	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ≠ ∅)
59	58	necomd 2997	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ∅ ≠ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
60	59	neneqd 2946	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
61	15, 26, 60	chvarfv 2234	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
62	61	intnand 490	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
63	62	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
64		opeq1 4874	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑁 → ⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩)
65		rdgeq2 8412	. . . . . . . . . . . . . . . . . . . 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 6899	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑁 → (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
69	68	eqeq2d 2744	. . . . . . . . . . . . . . . . 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 36266	. . . . . . . . . . . . . . 15 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}
73	71, 72	eqtr4di 2791	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = (𝑈↑↑𝑁))
74	73	eleq2d 2820	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ 𝑦 ∈ (𝑈↑↑𝑁)))
75		abid 2714	. . . . . . . . . . . . 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 482	. . . . . . . . . . 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 414	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
80	4, 79	biimtrid 241	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
81	80	expdimp 454	. . . . . . 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 3989	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈ 𝑛)) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
84	83	ex 414	. . . 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 3546	. 2 ⊢ (𝑁 ∈ ω → ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
87	86	anabsi5 668	1 ⊢ ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   ≠ wne 2941  Vcvv 3475   ⊆ wss 3949  ∅c0 4323  ifcif 4529  ⟨cop 4635  ∪ cuni 4909   × cxp 5675  Oncon0 6365  suc csuc 6367  ‘cfv 6544   ∈ cmpo 7411  ωcom 7855  1^st c1st 7973  reccrdg 8409  1_oc1o 8459  ↑↑cfinxp 36264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-finxp 36265

This theorem is referenced by:  finxpsuclem  36278