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Definition df-finxp 37366
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 8936 or df-map 8866 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 5148 can be used on it, and df-fv 6570 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 37374 and finxpsuc 37380 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 37365 . 2 class (𝑈↑↑𝑁)
4 com 7886 . . . . 5 class ω
52, 4wcel 2105 . . . 4 wff 𝑁 ∈ ω
6 c0 4338 . . . . 5 class
7 vn . . . . . . . 8 setvar 𝑛
8 vx . . . . . . . 8 setvar 𝑥
9 cvv 3477 . . . . . . . 8 class V
107cv 1535 . . . . . . . . . . 11 class 𝑛
11 c1o 8497 . . . . . . . . . . 11 class 1o
1210, 11wceq 1536 . . . . . . . . . 10 wff 𝑛 = 1o
138cv 1535 . . . . . . . . . . 11 class 𝑥
1413, 1wcel 2105 . . . . . . . . . 10 wff 𝑥𝑈
1512, 14wa 395 . . . . . . . . 9 wff (𝑛 = 1o𝑥𝑈)
169, 1cxp 5686 . . . . . . . . . . 11 class (V × 𝑈)
1713, 16wcel 2105 . . . . . . . . . 10 wff 𝑥 ∈ (V × 𝑈)
1810cuni 4911 . . . . . . . . . . 11 class 𝑛
19 c1st 8010 . . . . . . . . . . . 12 class 1st
2013, 19cfv 6562 . . . . . . . . . . 11 class (1st𝑥)
2118, 20cop 4636 . . . . . . . . . 10 class 𝑛, (1st𝑥)⟩
2210, 13cop 4636 . . . . . . . . . 10 class 𝑛, 𝑥
2317, 21, 22cif 4530 . . . . . . . . 9 class if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)
2415, 6, 23cif 4530 . . . . . . . 8 class if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
257, 8, 4, 9, 24cmpo 7432 . . . . . . 7 class (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
26 vy . . . . . . . . 9 setvar 𝑦
2726cv 1535 . . . . . . . 8 class 𝑦
282, 27cop 4636 . . . . . . 7 class 𝑁, 𝑦
2925, 28crdg 8447 . . . . . 6 class rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)
302, 29cfv 6562 . . . . 5 class (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
316, 30wceq 1536 . . . 4 wff ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
325, 31wa 395 . . 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 2711 . 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 1536 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  37367  finxpeq1  37368  finxpeq2  37369  csbfinxpg  37370  finxp0  37373  finxp1o  37374  finxpnom  37383
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