Definition df-vtx 28978

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 28976	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3437	. . 3 class V
4	2	cv 1540	. . . . 5 class 𝑔
5	3, 3	cxp 5617	. . . . 5 class (V × V)
6	4, 5	wcel 2113	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7925	. . . . 5 class 1st
8	4, 7	cfv 6486	. . . 4 class (1st ‘𝑔)
9		cbs 17122	. . . . 5 class Base
10	4, 9	cfv 6486	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4474	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5174	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1541	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  28980