Definition df-vtx 28932

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 28930	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3450	. . 3 class V
4	2	cv 1539	. . . . 5 class 𝑔
5	3, 3	cxp 5639	. . . . 5 class (V × V)
6	4, 5	wcel 2109	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7969	. . . . 5 class 1st
8	4, 7	cfv 6514	. . . 4 class (1st ‘𝑔)
9		cbs 17186	. . . . 5 class Base
10	4, 9	cfv 6514	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4491	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5191	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1540	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  28934