Definition df-k1 35993

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

Assertion

Ref	Expression
df-k1	⊢ 𝐾1 = {⟨𝑧, 𝑥⟩ ∣ (𝑥 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}

Distinct variable group:   𝑥,𝑛,𝑦,𝑧

Detailed syntax breakdown of Definition df-k1

Step	Hyp	Ref	Expression
1		ck1 35984	. 2 class 𝐾1
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1569	. . . . 5 class 𝑥
4		con0 6380	. . . . 5 class On
5	3, 4	wcel 2147	. . . 4 wff 𝑥 ∈ On
6		vz	. . . . . . . 8 setvar 𝑧
7	6	cv 1569	. . . . . . 7 class 𝑧
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1569	. . . . . . . . 9 class 𝑦
10		vn	. . . . . . . . . 10 setvar 𝑛
11	10	cv 1569	. . . . . . . . 9 class 𝑛
12	3, 9, 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 ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
17		c9o 35857	. . . . 5 class 9o
18	16, 10, 17	wrex 3090	. . . 4 wff ∃𝑛 ∈ 9o ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
19	5, 18	wa 401	. . 3 wff (𝑥 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))
20	19, 6, 2	copab 5170	. 2 class {⟨𝑧, 𝑥⟩ ∣ (𝑥 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}
21	1, 20	wceq 1570	1 wff 𝐾1 = {⟨𝑧, 𝑥⟩ ∣ (𝑥 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}

This definition is used by:  dfk12  36012