Definition df-finxp 36260

Description: Define Cartesian exponentiation on a class.

Note that this definition is limited to finite exponents, since it is defined using nested ordered pairs. If tuples of infinite length are needed, or if they might be needed in the future, use df-ixp 8891 or df-map 8821 instead. The main advantage of this definition is that it integrates better with functions and relations. For example if 𝑅 is a subset of (𝐴↑↑2_o), then df-br 5149 can be used on it, and df-fv 6551 can also be used, and so on.

It's also worth keeping in mind that ((𝑈↑↑𝑀) × (𝑈↑↑𝑁)) is generally not equal to (𝑈↑↑(𝑀 +_o 𝑁)).

This definition is technical. Use finxp1o 36268 and finxpsuc 36274 for a more standard recursive experience. (Contributed by ML, 16-Oct-2020.)

Assertion

Ref	Expression
df-finxp	⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}

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

Detailed syntax breakdown of Definition df-finxp

Step	Hyp	Ref	Expression
1		cU	. . 3 class 𝑈
2		cN	. . 3 class 𝑁
3	1, 2	cfinxp 36259	. 2 class (𝑈↑↑𝑁)
4		com 7854	. . . . 5 class ω
5	2, 4	wcel 2106	. . . 4 wff 𝑁 ∈ ω
6		c0 4322	. . . . 5 class ∅
7		vn	. . . . . . . 8 setvar 𝑛
8		vx	. . . . . . . 8 setvar 𝑥
9		cvv 3474	. . . . . . . 8 class V
10	7	cv 1540	. . . . . . . . . . 11 class 𝑛
11		c1o 8458	. . . . . . . . . . 11 class 1o
12	10, 11	wceq 1541	. . . . . . . . . 10 wff 𝑛 = 1o
13	8	cv 1540	. . . . . . . . . . 11 class 𝑥
14	13, 1	wcel 2106	. . . . . . . . . 10 wff 𝑥 ∈ 𝑈
15	12, 14	wa 396	. . . . . . . . 9 wff (𝑛 = 1o ∧ 𝑥 ∈ 𝑈)
16	9, 1	cxp 5674	. . . . . . . . . . 11 class (V × 𝑈)
17	13, 16	wcel 2106	. . . . . . . . . 10 wff 𝑥 ∈ (V × 𝑈)
18	10	cuni 4908	. . . . . . . . . . 11 class ∪ 𝑛
19		c1st 7972	. . . . . . . . . . . 12 class 1st
20	13, 19	cfv 6543	. . . . . . . . . . 11 class (1st ‘𝑥)
21	18, 20	cop 4634	. . . . . . . . . 10 class ⟨∪ 𝑛, (1st ‘𝑥)⟩
22	10, 13	cop 4634	. . . . . . . . . 10 class ⟨𝑛, 𝑥⟩
23	17, 21, 22	cif 4528	. . . . . . . . 9 class if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)
24	15, 6, 23	cif 4528	. . . . . . . 8 class if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
25	7, 8, 4, 9, 24	cmpo 7410	. . . . . . 7 class (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
26		vy	. . . . . . . . 9 setvar 𝑦
27	26	cv 1540	. . . . . . . 8 class 𝑦
28	2, 27	cop 4634	. . . . . . 7 class ⟨𝑁, 𝑦⟩
29	25, 28	crdg 8408	. . . . . 6 class rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)
30	2, 29	cfv 6543	. . . . 5 class (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
31	6, 30	wceq 1541	. . . 4 wff ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
32	5, 31	wa 396	. . 3 wff (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))
33	32, 26	cab 2709	. 2 class {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
34	3, 33	wceq 1541	1 wff (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}

This definition is referenced by:  dffinxpf  36261  finxpeq1  36262  finxpeq2  36263  csbfinxpg  36264  finxp0  36267  finxp1o  36268  finxpnom  36277