Theorem finxpreclem6 34671

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 2900	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ ω ↔ 𝑁 ∈ ω))
2		eleq2 2901	. . . . 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 471	. . . . . . . . 9 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))))
5		nfv 1911	. . . . . . . . . . . . . . 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 7229	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
8	6, 7	nfcxfr 2975	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐹
9		nfcv 2977	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥⟨𝑛, 𝑦⟩
10	8, 9	nfrdg 8044	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥rec(𝐹, ⟨𝑛, 𝑦⟩)
11		nfcv 2977	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑛
12	10, 11	nffv 6674	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
13	12	nfeq2 2995	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
14	13	nfn 1853	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)
15	5, 14	nfim 1893	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
16		eleq1 2900	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (V × 𝑈) ↔ 𝑦 ∈ (V × 𝑈)))
17	16	notbid 320	. . . . . . . . . . . . . . . . 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 4797	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ⟨𝑛, 𝑥⟩ = ⟨𝑛, 𝑦⟩)
21		rdgeq2 8042	. . . . . . . . . . . . . . . . . . 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 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
24	23	eqeq2d 2832	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
25	24	notbid 320	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ↔ ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
26	19, 25	imbi12d 347	. . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) ↔ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))))
28		vex 3497	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑛 ∈ V
29		fveqeq2 6673	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ∅ → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘∅) = ⟨𝑛, 𝑥⟩))
30		fveqeq2 6673	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩))
31		fveqeq2 6673	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = suc 𝑜 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = ⟨𝑛, 𝑥⟩))
32		opex 5348	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
33	32	rdg0 8051	. . . . . . . . . . . . . . . . . . . . . . . 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 7580	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑜 ∈ ω → 𝑜 ∈ On)
36		rdgsuc 8054	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6664	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩ → (𝐹‘(rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜)) = (𝐹‘⟨𝑛, 𝑥⟩))
39	37, 38	sylan9eq 2876	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑜 ∈ ω ∧ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑜) = ⟨𝑛, 𝑥⟩) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘suc 𝑜) = (𝐹‘⟨𝑛, 𝑥⟩))
40	6	finxpreclem5 34670	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))
41	40	imp 409	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
42	39, 41	sylan9eq 2876	. . . . . . . . . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . . 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 7604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
46		eleq1 2900	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ ω ↔ 𝑚 ∈ ω))
47		fveqeq2 6673	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → ((rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩ ↔ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑚) = ⟨𝑛, 𝑥⟩))
48	47	imbi2d 343	. . . . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
51	50	equcoms 2023	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)))
52	28, 51	vtocle 3583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ω → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
53	27, 52	syl5bir 245	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ω → ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩))
54	53	anabsi5 667	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) = ⟨𝑛, 𝑥⟩)
55		vex 3497	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
56	28, 55	opnzi 5358	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑛, 𝑥⟩ ≠ ∅
57	56	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ⟨𝑛, 𝑥⟩ ≠ ∅)
58	54, 57	eqnetrd 3083	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛) ≠ ∅)
59	58	necomd 3071	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ∅ ≠ (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
60	59	neneqd 3021	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑥 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑥⟩)‘𝑛))
61	15, 26, 60	chvarfv 2238	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))
62	61	intnand 491	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
63	62	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
64		abid 2803	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)))
65		opeq1 4796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑁 → ⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩)
66		rdgeq2 8042	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑛, 𝑦⟩ = ⟨𝑁, 𝑦⟩ → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩))
67	65, 66	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑁 → rec(𝐹, ⟨𝑛, 𝑦⟩) = rec(𝐹, ⟨𝑁, 𝑦⟩))
68		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
69	67, 68	fveq12d 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑁 → (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
70	69	eqeq2d 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑁 → (∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛) ↔ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁)))
71	1, 70	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))))
72	71	abbidv 2885	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))})
73	6	dffinxpf 34660	. . . . . . . . . . . . . . 15 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}
74	72, 73	syl6eqr 2874	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} = (𝑈↑↑𝑁))
75	74	eleq2d 2898	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑦 ∈ {𝑦 ∣ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))} ↔ 𝑦 ∈ (𝑈↑↑𝑁)))
76	64, 75	syl5rbbr 288	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
77	76	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → (𝑦 ∈ (𝑈↑↑𝑁) ↔ (𝑛 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑛, 𝑦⟩)‘𝑛))))
78	63, 77	mtbird 327	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈)))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁))
79	78	ex 415	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ (1o ∈ 𝑛 ∧ ¬ 𝑦 ∈ (V × 𝑈))) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
80	4, 79	syl5bi 244	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (((𝑛 ∈ ω ∧ 1o ∈ 𝑛) ∧ ¬ 𝑦 ∈ (V × 𝑈)) → ¬ 𝑦 ∈ (𝑈↑↑𝑁)))
81	80	expdimp 455	. . . . . . 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 3972	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ (𝑛 ∈ ω ∧ 1o ∈ 𝑛)) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))
84	83	ex 415	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
85	3, 84	sylbird 262	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
86	85	vtocleg 3580	. 2 ⊢ (𝑁 ∈ ω → ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)))
87	86	anabsi5 667	1 ⊢ ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799   ≠ wne 3016  Vcvv 3494   ⊆ wss 3935  ∅c0 4290  ifcif 4466  ⟨cop 4566  ∪ cuni 4831   × cxp 5547  Oncon0 6185  suc csuc 6187  ‘cfv 6349   ∈ cmpo 7152  ωcom 7574  1^st c1st 7681  reccrdg 8039  1_oc1o 8089  ↑↑cfinxp 34658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-finxp 34659

This theorem is referenced by:  finxpsuclem  34672