Definition df-vtx 29033

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 29031	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3488	. . 3 class V
4	2	cv 1536	. . . . 5 class 𝑔
5	3, 3	cxp 5698	. . . . 5 class (V × V)
6	4, 5	wcel 2108	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 8028	. . . . 5 class 1st
8	4, 7	cfv 6573	. . . 4 class (1st ‘𝑔)
9		cbs 17258	. . . . 5 class Base
10	4, 9	cfv 6573	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4548	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5249	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1537	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  29035