Definition df-vtx 29558

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 29556	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3451	. . 3 class V
4	2	cv 1569	. . . . 5 class 𝑔
5	3, 3	cxp 5649	. . . . 5 class (V × V)
6	4, 5	wcel 2145	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7988	. . . . 5 class 1st
8	4, 7	cfv 6531	. . . 4 class (1st ‘𝑔)
9		cbs 17367	. . . . 5 class Base
10	4, 9	cfv 6531	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4482	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5186	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1570	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is used by:  vtxval  29560