Definition df-k2 35994

Description: Define a function that takes an ordinal and returns the second argument of the ordered triple ⟨𝑥, 𝑦, 𝑛⟩ such that the ordinal equals (𝐽‘⟨𝑥, 𝑦, 𝑛⟩). Based on the second case of Definition 15.7 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 2-Sep-2026.)

Assertion

Ref	Expression
df-k2	⊢ 𝐾2 = {⟨𝑧, 𝑦⟩ ∣ (𝑦 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑥 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}

Distinct variable group:   𝑥,𝑛,𝑦,𝑧

Detailed syntax breakdown of Definition df-k2

Step	Hyp	Ref	Expression
1		ck2 35985	. 2 class 𝐾2
2		vy	. . . . . 6 setvar 𝑦
3	2	cv 1569	. . . . 5 class 𝑦
4		con0 6380	. . . . 5 class On
5	3, 4	wcel 2147	. . . 4 wff 𝑦 ∈ On
6		vz	. . . . . . . 8 setvar 𝑧
7	6	cv 1569	. . . . . . 7 class 𝑧
8		vx	. . . . . . . . . 10 setvar 𝑥
9	8	cv 1569	. . . . . . . . 9 class 𝑥
10		vn	. . . . . . . . . 10 setvar 𝑛
11	10	cv 1569	. . . . . . . . 9 class 𝑛
12	9, 3, 11	cotp 4861	. . . . . . . 8 class ⟨𝑥, 𝑦, 𝑛⟩
13		cj 35983	. . . . . . . 8 class 𝐽
14	12, 13	cfv 6558	. . . . . . 7 class (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
15	7, 14	wceq 1570	. . . . . 6 wff 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
16	15, 8, 4	wrex 3090	. . . . 5 wff ∃𝑥 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
17		c9o 35857	. . . . 5 class 9o
18	16, 10, 17	wrex 3090	. . . 4 wff ∃𝑛 ∈ 9o ∃𝑥 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
19	5, 18	wa 401	. . 3 wff (𝑦 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑥 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))
20	19, 6, 2	copab 5170	. 2 class {⟨𝑧, 𝑦⟩ ∣ (𝑦 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑥 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}
21	1, 20	wceq 1570	1 wff 𝐾2 = {⟨𝑧, 𝑦⟩ ∣ (𝑦 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑥 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}

This definition is used by:  dfk22  36013