MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-oexp Structured version   Visualization version   GIF version

Definition df-oexp 7511
Description: Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)
Assertion
Ref Expression
df-oexp 𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-oexp
StepHypRef Expression
1 coe 7504 . 2 class 𝑜
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 5682 . . 3 class On
52cv 1479 . . . . 5 class 𝑥
6 c0 3891 . . . . 5 class
75, 6wceq 1480 . . . 4 wff 𝑥 = ∅
8 c1o 7498 . . . . 5 class 1𝑜
93cv 1479 . . . . 5 class 𝑦
108, 9cdif 3552 . . . 4 class (1𝑜𝑦)
11 vz . . . . . . 7 setvar 𝑧
12 cvv 3186 . . . . . . 7 class V
1311cv 1479 . . . . . . . 8 class 𝑧
14 comu 7503 . . . . . . . 8 class ·𝑜
1513, 5, 14co 6604 . . . . . . 7 class (𝑧 ·𝑜 𝑥)
1611, 12, 15cmpt 4673 . . . . . 6 class (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥))
1716, 8crdg 7450 . . . . 5 class rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)
189, 17cfv 5847 . . . 4 class (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)
197, 10, 18cif 4058 . . 3 class if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))
202, 3, 4, 4, 19cmpt2 6606 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)))
211, 20wceq 1480 1 wff 𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)))
Colors of variables: wff setvar class
This definition is referenced by:  fnoe  7535  oev  7539
  Copyright terms: Public domain W3C validator