Definition df-vtx 28762

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 28760	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3468	. . 3 class V
4	2	cv 1532	. . . . 5 class 𝑔
5	3, 3	cxp 5667	. . . . 5 class (V × V)
6	4, 5	wcel 2098	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7969	. . . . 5 class 1st
8	4, 7	cfv 6536	. . . 4 class (1st ‘𝑔)
9		cbs 17151	. . . . 5 class Base
10	4, 9	cfv 6536	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4523	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5224	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1533	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  28764