Definition df-se 4056

Description: Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)

Assertion

Ref	Expression
df-se	⊢ (𝑅 Se A ↔ ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V)

Distinct variable groups:   x,y,𝑅   x,A,y

Detailed syntax breakdown of Definition df-se

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cR	. . 3 class 𝑅
3	1, 2	wse 4055	. 2 wff 𝑅 Se A
4		vy	. . . . . . 7 setvar y
5	4	cv 1241	. . . . . 6 class y
6		vx	. . . . . . 7 setvar x
7	6	cv 1241	. . . . . 6 class x
8	5, 7, 2	wbr 3755	. . . . 5 wff y𝑅x
9	8, 4, 1	crab 2304	. . . 4 class {y ∈ A ∣ y𝑅x}
10		cvv 2551	. . . 4 class V
11	9, 10	wcel 1390	. . 3 wff {y ∈ A ∣ y𝑅x} ∈ V
12	11, 6, 1	wral 2300	. 2 wff ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V
13	3, 12	wb 98	1 wff (𝑅 Se A ↔ ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V)

This definition is referenced by:  seex  4057  exse  4058  sess1  4059  sess2  4060  nfse  4063  epse  4064  seinxp  4354  dfse2  4641  exse2  4642