Definition df-se 5570

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 5567	. 2 wff 𝑅 Se 𝐴
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1540	. . . . . 6 class 𝑦
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1540	. . . . . 6 class 𝑥
8	5, 7, 2	wbr 5091	. . . . 5 wff 𝑦𝑅𝑥
9	8, 4, 1	crab 3395	. . . 4 class {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥}
10		cvv 3436	. . . 4 class V
11	9, 10	wcel 2111	. . 3 wff {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V
12	11, 6, 1	wral 3047	. 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  5575  exse  5576  sess1  5581  sess2  5582  nfse  5590  epse  5598  seinxp  5700  dfse2  6049  exse2  7847  lrrecse  27886  fineqvnttrclse  35142  weiunse  36508  bj-seex  36962