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 28824
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 28822 . 2 class Vtx
2 vg . . 3 setvar 𝑔
3 cvv 3471 . . 3 class V
42cv 1533 . . . . 5 class 𝑔
53, 3cxp 5676 . . . . 5 class (V × V)
64, 5wcel 2099 . . . 4 wff 𝑔 ∈ (V × V)
7 c1st 7991 . . . . 5 class 1st
84, 7cfv 6548 . . . 4 class (1st𝑔)
9 cbs 17180 . . . . 5 class Base
104, 9cfv 6548 . . . 4 class (Base‘𝑔)
116, 8, 10cif 4529 . . 3 class if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔))
122, 3, 11cmpt 5231 . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
131, 12wceq 1534 1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
Colors of variables: wff setvar class
This definition is referenced by:  vtxval  28826
  Copyright terms: Public domain W3C validator