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

Definition df-vtx 26096
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 26094 . 2 class Vtx
2 vg . . 3 setvar 𝑔
3 cvv 3340 . . 3 class V
42cv 1631 . . . . 5 class 𝑔
53, 3cxp 5264 . . . . 5 class (V × V)
64, 5wcel 2139 . . . 4 wff 𝑔 ∈ (V × V)
7 c1st 7332 . . . . 5 class 1st
84, 7cfv 6049 . . . 4 class (1st𝑔)
9 cbs 16079 . . . . 5 class Base
104, 9cfv 6049 . . . 4 class (Base‘𝑔)
116, 8, 10cif 4230 . . 3 class if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔))
122, 3, 11cmpt 4881 . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
131, 12wceq 1632 1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
Colors of variables: wff setvar class
This definition is referenced by:  vtxval  26098  vtxvalOLD  26100
  Copyright terms: Public domain W3C validator