Definition df-finxp 37875

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 8880 or df-map 8810 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 5101 can be used on it, and df-fv 6529 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 37883 and finxpsuc 37889 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 37874	. 2 class (𝑈↑↑𝑁)
4		com 7846	. . . . 5 class ω
5	2, 4	wcel 2142	. . . 4 wff 𝑁 ∈ ω
6		c0 4285	. . . . 5 class ∅
7		vn	. . . . . . . 8 setvar 𝑛
8		vx	. . . . . . . 8 setvar 𝑥
9		cvv 3454	. . . . . . . 8 class V
10	7	cv 1559	. . . . . . . . . . 11 class 𝑛
11		c1o 8430	. . . . . . . . . . 11 class 1o
12	10, 11	wceq 1560	. . . . . . . . . 10 wff 𝑛 = 1o
13	8	cv 1559	. . . . . . . . . . 11 class 𝑥
14	13, 1	wcel 2142	. . . . . . . . . 10 wff 𝑥 ∈ 𝑈
15	12, 14	wa 399	. . . . . . . . 9 wff (𝑛 = 1o ∧ 𝑥 ∈ 𝑈)
16	9, 1	cxp 5645	. . . . . . . . . . 11 class (V × 𝑈)
17	13, 16	wcel 2142	. . . . . . . . . 10 wff 𝑥 ∈ (V × 𝑈)
18	10	cuni 4865	. . . . . . . . . . 11 class ∪ 𝑛
19		c1st 7968	. . . . . . . . . . . 12 class 1st
20	13, 19	cfv 6521	. . . . . . . . . . 11 class (1st ‘𝑥)
21	18, 20	cop 4588	. . . . . . . . . 10 class ⟨∪ 𝑛, (1st ‘𝑥)⟩
22	10, 13	cop 4588	. . . . . . . . . 10 class ⟨𝑛, 𝑥⟩
23	17, 21, 22	cif 4480	. . . . . . . . 9 class if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)
24	15, 6, 23	cif 4480	. . . . . . . 8 class if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
25	7, 8, 4, 9, 24	cmpo 7398	. . . . . . 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 1559	. . . . . . . 8 class 𝑦
28	2, 27	cop 4588	. . . . . . 7 class ⟨𝑁, 𝑦⟩
29	25, 28	crdg 8380	. . . . . 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 6521	. . . . 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 1560	. . . 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 399	. . 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 2740	. 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 1560	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  37876  finxpeq1  37877  finxpeq2  37878  csbfinxpg  37879  finxp0  37882  finxp1o  37883  finxpnom  37892