Definition df-dcin 13001

Description: Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)

Assertion

Ref	Expression
df-dcin	⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Detailed syntax breakdown of Definition df-dcin

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	wdcin 13000	. 2 wff 𝐴 DECIDin 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1330	. . . . 5 class 𝑥
6	5, 1	wcel 1480	. . . 4 wff 𝑥 ∈ 𝐴
7	6	wdc 819	. . 3 wff DECID 𝑥 ∈ 𝐴
8	7, 4, 2	wral 2416	. 2 wff ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴
9	3, 8	wb 104	1 wff (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)

This definition is referenced by:  decidi  13002  decidr  13003  sumdc2  13006