Definition df-vtx 29092

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 29090	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3432	. . 3 class V
4	2	cv 1546	. . . . 5 class 𝑔
5	3, 3	cxp 5623	. . . . 5 class (V × V)
6	4, 5	wcel 2119	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7936	. . . . 5 class 1st
8	4, 7	cfv 6492	. . . 4 class (1st ‘𝑔)
9		cbs 17177	. . . . 5 class Base
10	4, 9	cfv 6492	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4461	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5160	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1547	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  29094