Definition df-vtx 29029

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), (1st ‘𝑔), (Base‘𝑔)))

Detailed syntax breakdown of Definition df-vtx

Step	Hyp	Ref	Expression
1		cvtx 29027	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3477	. . 3 class V
4	2	cv 1535	. . . . 5 class 𝑔
5	3, 3	cxp 5686	. . . . 5 class (V × V)
6	4, 5	wcel 2105	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 8010	. . . . 5 class 1st
8	4, 7	cfv 6562	. . . 4 class (1st ‘𝑔)
9		cbs 17244	. . . . 5 class Base
10	4, 9	cfv 6562	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4530	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5230	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1536	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  29031