Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-vtx Structured version   Visualization version   GIF version

Definition df-vtx 40212
Description: Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)
Assertion
Ref Expression
df-vtx Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))

Detailed syntax breakdown of Definition df-vtx
StepHypRef Expression
1 cvtx 40210 . 2 class Vtx
2 vg . . 3 setvar 𝑔
3 cvv 3172 . . 3 class V
42cv 1473 . . . . 5 class 𝑔
53, 3cxp 5025 . . . . 5 class (V × V)
64, 5wcel 1976 . . . 4 wff 𝑔 ∈ (V × V)
7 c1st 7034 . . . . 5 class 1st
84, 7cfv 5789 . . . 4 class (1st𝑔)
9 cbs 15643 . . . . 5 class Base
104, 9cfv 5789 . . . 4 class (Base‘𝑔)
116, 8, 10cif 4035 . . 3 class if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔))
122, 3, 11cmpt 4637 . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
131, 12wceq 1474 1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
Colors of variables: wff setvar class
This definition is referenced by:  vtxval  40214
  Copyright terms: Public domain W3C validator