Definition df-ind 9294

Description: Define the indicator function generator. It generates an indicator function ((𝟭‘𝑂)‘𝐴) for a given domain 𝑂 and a given subset 𝐴 of the domain, see indval 9296. In contrast to the definitions and notations in Wikipedia and [Lang2] p. 3, the domain and the subset are always mentioned explicitly. (Contributed by Thierry Arnoux, 20-Jan-2017.)

Assertion

Ref	Expression
df-ind	⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))

Distinct variable group:   𝑜,𝑎,𝑥

Detailed syntax breakdown of Definition df-ind

Step	Hyp	Ref	Expression
1		cind 9293	. 2 class 𝟭
2		vo	. . 3 setvar 𝑜
3		cvv 2821	. . 3 class V
4		va	. . . 4 setvar 𝑎
5	2	cv 1401	. . . . 5 class 𝑜
6	5	cpw 3688	. . . 4 class 𝒫 𝑜
7		vx	. . . . 5 setvar 𝑥
8	7, 4	wel 2210	. . . . . 6 wff 𝑥 ∈ 𝑎
9		c1 8180	. . . . . 6 class 1
10		cc0 8179	. . . . . 6 class 0
11	8, 9, 10	cif 3638	. . . . 5 class if(𝑥 ∈ 𝑎, 1, 0)
12	7, 5, 11	cmpt 4192	. . . 4 class (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))
13	4, 6, 12	cmpt 4192	. . 3 class (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0)))
14	2, 3, 13	cmpt 4192	. 2 class (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
15	1, 14	wceq 1402	1 wff 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))

This definition is used by:  indv  9295  indval0  9297