Definition df-se 5585

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 5582	. 2 wff 𝑅 Se 𝐴
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1541	. . . . . 6 class 𝑦
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1541	. . . . . 6 class 𝑥
8	5, 7, 2	wbr 5085	. . . . 5 wff 𝑦𝑅𝑥
9	8, 4, 1	crab 3389	. . . 4 class {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥}
10		cvv 3429	. . . 4 class V
11	9, 10	wcel 2114	. . 3 wff {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V
12	11, 6, 1	wral 3051	. 2 wff ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V
13	3, 12	wb 206	1 wff (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)

This definition is referenced by:  seex  5590  exse  5591  sess1  5596  sess2  5597  nfse  5605  epse  5613  seinxp  5715  dfse2  6065  exse2  7868  lrrecse  27934  fineqvnttrclse  35268  weiunse  36650  bj-seex  37229