Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-se Structured version   Visualization version   GIF version

Definition df-se 4998
 Description: Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
Assertion
Ref Expression
df-se (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝐴,𝑦

Detailed syntax breakdown of Definition df-se
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cR . . 3 class 𝑅
31, 2wse 4995 . 2 wff 𝑅 Se 𝐴
4 vy . . . . . . 7 setvar 𝑦
54cv 1474 . . . . . 6 class 𝑦
6 vx . . . . . . 7 setvar 𝑥
76cv 1474 . . . . . 6 class 𝑥
85, 7, 2wbr 4583 . . . . 5 wff 𝑦𝑅𝑥
98, 4, 1crab 2900 . . . 4 class {𝑦𝐴𝑦𝑅𝑥}
10 cvv 3173 . . . 4 class V
119, 10wcel 1977 . . 3 wff {𝑦𝐴𝑦𝑅𝑥} ∈ V
1211, 6, 1wral 2896 . 2 wff 𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V
133, 12wb 195 1 wff (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
 Colors of variables: wff setvar class This definition is referenced by:  seex  5001  exse  5002  sess1  5006  sess2  5007  nfse  5013  epse  5021  seinxp  5108  dfse2  5418  exse2  6998  bj-seex  32111
 Copyright terms: Public domain W3C validator