Definition df-vtx 27366

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 27364	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3431	. . 3 class V
4	2	cv 1541	. . . . 5 class 𝑔
5	3, 3	cxp 5588	. . . . 5 class (V × V)
6	4, 5	wcel 2110	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7822	. . . . 5 class 1st
8	4, 7	cfv 6432	. . . 4 class (1st ‘𝑔)
9		cbs 16910	. . . . 5 class Base
10	4, 9	cfv 6432	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4465	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5162	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1542	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  27368