Definition df-vtx 25675

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 25673	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3173	. . 3 class V
4	2	cv 1474	. . . . 5 class 𝑔
5	3, 3	cxp 5036	. . . . 5 class (V × V)
6	4, 5	wcel 1977	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7057	. . . . 5 class 1st
8	4, 7	cfv 5804	. . . 4 class (1st ‘𝑔)
9		cbs 15695	. . . . 5 class Base
10	4, 9	cfv 5804	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4036	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4643	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1475	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  25677