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Definition df-finxp 34647
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 8454 or df-map 8400 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↑↑2o), then df-br 5058 can be used on it, and df-fv 6356 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 34655 and finxpsuc 34661 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 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
Distinct variable groups:   𝑈,𝑛,𝑥,𝑦   𝑛,𝑁,𝑥,𝑦

Detailed syntax breakdown of Definition df-finxp
StepHypRef Expression
1 cU . . 3 class 𝑈
2 cN . . 3 class 𝑁
31, 2cfinxp 34646 . 2 class (𝑈↑↑𝑁)
4 com 7572 . . . . 5 class ω
52, 4wcel 2107 . . . 4 wff 𝑁 ∈ ω
6 c0 4289 . . . . 5 class
7 vn . . . . . . . 8 setvar 𝑛
8 vx . . . . . . . 8 setvar 𝑥
9 cvv 3493 . . . . . . . 8 class V
107cv 1529 . . . . . . . . . . 11 class 𝑛
11 c1o 8087 . . . . . . . . . . 11 class 1o
1210, 11wceq 1530 . . . . . . . . . 10 wff 𝑛 = 1o
138cv 1529 . . . . . . . . . . 11 class 𝑥
1413, 1wcel 2107 . . . . . . . . . 10 wff 𝑥𝑈
1512, 14wa 398 . . . . . . . . 9 wff (𝑛 = 1o𝑥𝑈)
169, 1cxp 5546 . . . . . . . . . . 11 class (V × 𝑈)
1713, 16wcel 2107 . . . . . . . . . 10 wff 𝑥 ∈ (V × 𝑈)
1810cuni 4830 . . . . . . . . . . 11 class 𝑛
19 c1st 7679 . . . . . . . . . . . 12 class 1st
2013, 19cfv 6348 . . . . . . . . . . 11 class (1st𝑥)
2118, 20cop 4565 . . . . . . . . . 10 class 𝑛, (1st𝑥)⟩
2210, 13cop 4565 . . . . . . . . . 10 class 𝑛, 𝑥
2317, 21, 22cif 4465 . . . . . . . . 9 class if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)
2415, 6, 23cif 4465 . . . . . . . 8 class if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
257, 8, 4, 9, 24cmpo 7150 . . . . . . 7 class (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
26 vy . . . . . . . . 9 setvar 𝑦
2726cv 1529 . . . . . . . 8 class 𝑦
282, 27cop 4565 . . . . . . 7 class 𝑁, 𝑦
2925, 28crdg 8037 . . . . . 6 class rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)
302, 29cfv 6348 . . . . 5 class (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
316, 30wceq 1530 . . . 4 wff ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
325, 31wa 398 . . 3 wff (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))
3332, 26cab 2797 . 2 class {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
343, 33wceq 1530 1 wff (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
Colors of variables: wff setvar class
This definition is referenced by:  dffinxpf  34648  finxpeq1  34649  finxpeq2  34650  csbfinxpg  34651  finxp0  34654  finxp1o  34655  finxpnom  34664
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