Definition df-vtx 29326

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 29324	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3455	. . 3 class V
4	2	cv 1569	. . . . 5 class 𝑔
5	3, 3	cxp 5661	. . . . 5 class (V × V)
6	4, 5	wcel 2143	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7985	. . . . 5 class 1st
8	4, 7	cfv 6538	. . . 4 class (1st ‘𝑔)
9		cbs 17270	. . . . 5 class Base
10	4, 9	cfv 6538	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4488	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5193	. 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 referenced by:  vtxval  29328