Definition df-vtx 26791

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 26789	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3441	. . 3 class V
4	2	cv 1537	. . . . 5 class 𝑔
5	3, 3	cxp 5517	. . . . 5 class (V × V)
6	4, 5	wcel 2111	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7669	. . . . 5 class 1st
8	4, 7	cfv 6324	. . . 4 class (1st ‘𝑔)
9		cbs 16475	. . . . 5 class Base
10	4, 9	cfv 6324	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4425	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5110	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1538	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  26793