Definition df-vtx 28883

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 28881	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3461	. . 3 class V
4	2	cv 1532	. . . . 5 class 𝑔
5	3, 3	cxp 5676	. . . . 5 class (V × V)
6	4, 5	wcel 2098	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7992	. . . . 5 class 1st
8	4, 7	cfv 6549	. . . 4 class (1st ‘𝑔)
9		cbs 17183	. . . . 5 class Base
10	4, 9	cfv 6549	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4530	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5232	. 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  28885