Definition df-finxp 36896

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 8923 or df-map 8853 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 5153 can be used on it, and df-fv 6561 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 36904 and finxpsuc 36910 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 36895	. 2 class (𝑈↑↑𝑁)
4		com 7876	. . . . 5 class ω
5	2, 4	wcel 2098	. . . 4 wff 𝑁 ∈ ω
6		c0 4326	. . . . 5 class ∅
7		vn	. . . . . . . 8 setvar 𝑛
8		vx	. . . . . . . 8 setvar 𝑥
9		cvv 3473	. . . . . . . 8 class V
10	7	cv 1532	. . . . . . . . . . 11 class 𝑛
11		c1o 8486	. . . . . . . . . . 11 class 1o
12	10, 11	wceq 1533	. . . . . . . . . 10 wff 𝑛 = 1o
13	8	cv 1532	. . . . . . . . . . 11 class 𝑥
14	13, 1	wcel 2098	. . . . . . . . . 10 wff 𝑥 ∈ 𝑈
15	12, 14	wa 394	. . . . . . . . 9 wff (𝑛 = 1o ∧ 𝑥 ∈ 𝑈)
16	9, 1	cxp 5680	. . . . . . . . . . 11 class (V × 𝑈)
17	13, 16	wcel 2098	. . . . . . . . . 10 wff 𝑥 ∈ (V × 𝑈)
18	10	cuni 4912	. . . . . . . . . . 11 class ∪ 𝑛
19		c1st 7997	. . . . . . . . . . . 12 class 1st
20	13, 19	cfv 6553	. . . . . . . . . . 11 class (1st ‘𝑥)
21	18, 20	cop 4638	. . . . . . . . . 10 class ⟨∪ 𝑛, (1st ‘𝑥)⟩
22	10, 13	cop 4638	. . . . . . . . . 10 class ⟨𝑛, 𝑥⟩
23	17, 21, 22	cif 4532	. . . . . . . . 9 class if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)
24	15, 6, 23	cif 4532	. . . . . . . 8 class if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
25	7, 8, 4, 9, 24	cmpo 7428	. . . . . . 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 1532	. . . . . . . 8 class 𝑦
28	2, 27	cop 4638	. . . . . . 7 class ⟨𝑁, 𝑦⟩
29	25, 28	crdg 8436	. . . . . 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 6553	. . . . 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 1533	. . . 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 394	. . 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 2705	. 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 1533	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  36897  finxpeq1  36898  finxpeq2  36899  csbfinxpg  36900  finxp0  36903  finxp1o  36904  finxpnom  36913