Definition df-bj-sngl 35083

Description: Definition of "singletonization". The class sngl 𝐴 is isomorphic to 𝐴 and since it contains only singletons, it can be easily be adjoined disjoint elements, which can be useful in various constructions. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
df-bj-sngl	⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}

Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-bj-sngl

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	bj-csngl 35082	. 2 class sngl 𝐴
3		vx	. . . . . 6 setvar 𝑥
4	3	cv 1538	. . . . 5 class 𝑥
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1538	. . . . . 6 class 𝑦
7	6	csn 4558	. . . . 5 class {𝑦}
8	4, 7	wceq 1539	. . . 4 wff 𝑥 = {𝑦}
9	8, 5, 1	wrex 3064	. . 3 wff ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}
10	9, 3	cab 2715	. 2 class {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}
11	2, 10	wceq 1539	1 wff sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}

This definition is referenced by:  bj-sngleq  35084  bj-elsngl  35085