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Definition df-finxp 36568
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 8894 or df-map 8824 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 6550 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 36576 and finxpsuc 36582 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 36567 . 2 class (𝑈↑↑𝑁)
4 com 7857 . . . . 5 class ω
52, 4wcel 2104 . . . 4 wff 𝑁 ∈ ω
6 c0 4321 . . . . 5 class
7 vn . . . . . . . 8 setvar 𝑛
8 vx . . . . . . . 8 setvar 𝑥
9 cvv 3472 . . . . . . . 8 class V
107cv 1538 . . . . . . . . . . 11 class 𝑛
11 c1o 8461 . . . . . . . . . . 11 class 1o
1210, 11wceq 1539 . . . . . . . . . 10 wff 𝑛 = 1o
138cv 1538 . . . . . . . . . . 11 class 𝑥
1413, 1wcel 2104 . . . . . . . . . 10 wff 𝑥𝑈
1512, 14wa 394 . . . . . . . . 9 wff (𝑛 = 1o𝑥𝑈)
169, 1cxp 5673 . . . . . . . . . . 11 class (V × 𝑈)
1713, 16wcel 2104 . . . . . . . . . 10 wff 𝑥 ∈ (V × 𝑈)
1810cuni 4907 . . . . . . . . . . 11 class 𝑛
19 c1st 7975 . . . . . . . . . . . 12 class 1st
2013, 19cfv 6542 . . . . . . . . . . 11 class (1st𝑥)
2118, 20cop 4633 . . . . . . . . . 10 class 𝑛, (1st𝑥)⟩
2210, 13cop 4633 . . . . . . . . . 10 class 𝑛, 𝑥
2317, 21, 22cif 4527 . . . . . . . . 9 class if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)
2415, 6, 23cif 4527 . . . . . . . 8 class if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
257, 8, 4, 9, 24cmpo 7413 . . . . . . 7 class (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
26 vy . . . . . . . . 9 setvar 𝑦
2726cv 1538 . . . . . . . 8 class 𝑦
282, 27cop 4633 . . . . . . 7 class 𝑁, 𝑦
2925, 28crdg 8411 . . . . . 6 class rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)
302, 29cfv 6542 . . . . 5 class (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
316, 30wceq 1539 . . . 4 wff ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
325, 31wa 394 . . 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 2707 . 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 1539 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  36569  finxpeq1  36570  finxpeq2  36571  csbfinxpg  36572  finxp0  36575  finxp1o  36576  finxpnom  36585
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