Definition df-vtx 29071

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 29069	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3440	. . 3 class V
4	2	cv 1540	. . . . 5 class 𝑔
5	3, 3	cxp 5622	. . . . 5 class (V × V)
6	4, 5	wcel 2113	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7931	. . . . 5 class 1st
8	4, 7	cfv 6492	. . . 4 class (1st ‘𝑔)
9		cbs 17136	. . . . 5 class Base
10	4, 9	cfv 6492	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4479	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5179	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1541	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  29073